Calculation of Roundabouts
Raffaele Mauro
Calculation of Roundabouts Capacity, Waiting Phenomena and Reliability
123
Prof. Raffaele Mauro Università Trento Dip.to Ingegneria Meccanica e Strutturale (DIMS) Via Mesiano, 77 38050 Trento Italy
[email protected]
ISBN 978-3-642-04550-9 e-ISBN 978-3-642-04551-6 DOI 10.1007/978-3-642-04551-6 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009939930 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my Mother, Invigilans Lucerna
Preface
Over the last decades, roundabouts have been increasingly used in building new atgrade intersections and in changing the layout of existing intersections. Therefore, we decided that it would be useful to collect, in one volume, the methods and procedures used to evaluate the operating conditions of this type of intersection. The organization of this book is as follows: Chapter 1 deals with the definitions of capacity, capacity indices for roundabouts, and parameters linked to waiting phenomena at entries, i.e., delays and queue lengths. This is preceded by an introduction to the fundamental concepts associated with statistical equilibrium and steady-state conditions. These general concepts of the theory of systems and control are applied to roundabouts. Chapter 2 starts with some examples of capacity formulas, selected from the three types that are available in today’s scientific and technical literature. Then, some criteria for taking into account the effects of pedestrian flows on entry and exit capacities are presented. The remaining part of this chapter is dedicated to the calculation procedures for flows entering the roundabout, to the capacity in case of saturation or oversaturation of one or more entries, and to simple capacity and total capacity. Chapter 3 covers the analysis of waiting phenomena under steady-state and transient conditions. The material in Chap. 3 includes the standard key results of simple probabilistic (Markovian) and deterministic queuing systems, as well as the results of some special time-dependent solutions for waiting phenomena. Chapter 3 has a more general scope because it presents results that may be used both for roundabouts and for any at-grade intersection. Chapter 4 deals with the application of the results and methods discussed in the previous chapters for the evaluation of waiting times, queue lengths, and levels of service of roundabouts. The calculation procedures illustrated are meant for operating conditions characterized by undersaturated, saturated, or over-saturated entries. Chapter 4 ends with the determination of the level of service of a roundabout.
vii
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Preface
All the chapters include worked examples that have been developed in sufficient detail to explain as clearly as possible the formulas and the procedures presented in the book. We do hope that the expository format that we have used, which is characterized by a plain style and supported by worked examples, will help the reader to easily understand and be able to use the materials discussed in the book. We also hope that this layout will help the highway traffic engineers analyze the operating conditions of roundabouts. Chapter 5 presents criteria to evaluate roundabout performance reliability. After introducing and justifying the adoption of reserve of capacity and rate of capacity as performance functions, the discussion is developed using a general calculation criterion in which the values that are involved in the limit state service condition – traffic demand and entry capacity – are random variables described by their probability density functions, that is to say by their distribution functions. A lower level criterion is then identified with which, on the basis of the estimation of suitable statistics of the performance function, a reliability index is calculated that can be compared to a prefixed reference value. Trento, Italy
Raffaele Mauro
Acknowledgments
My continuing interest in the study of roundabouts is due to my mentor at University of Naples “Federico II”, professor Tommaso Esposito, who, over ten years ago, urged me to publish the first two works that appeared in Italy on this type of intersection. Most of this book is the product of our common studies and consulting work. I have often discussed with Giulio Erberto Cantarella, professor of transportation system theory at Salerno University, about steady-state conditions, transient states, and time-dependent solutions, and he gave me many fruitful suggestions and helped me focus my attention on interesting and important aspects of the topic. Special thanks also to Michele Corradini and Alessandro Romania. Michele Corradini and Alessandro Romania, both of whom have M.S. degrees in Civil Engineering and who were my students at Trento University, designed figures, checked analytical developments, verified the accuracy of numerical calculations, and offered numerous, relevant observations. This edition is based on an English translation prepared by Dr. Savino Carrella. He also has patiently revised, corrected, and improved the manuscript. Finally, I wish to thank the Province of Trento that has supported with a grant my recent researches on roundabouts (Special Project CRS – Performance analysis of roundabouts).
ix
Contents
1 Calculation of Roundabouts: Problem Definition . . . . . . . . . . 1.1 Statistical Equilibrium and Steady-State Conditions . . . . . . . 1.1.1 A Worked Example on Traffic Demand at a Roundabout 1.2 Capacity and Capacity Indices . . . . . . . . . . . . . . . . . . 1.3 Delays and Queue Lengths . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
1 2 4 7 9 13
2 Capacity Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Capacity Calculation at Steady-State Conditions . . . . . . . . 2.1.1 Brilon-Bondzio Formula (Germany) . . . . . . . . . . 2.1.2 Bovy et al. Formula (Switzerland) . . . . . . . . . . . 2.1.3 TRRL Formula (United Kingdom) . . . . . . . . . . . 2.1.4 GIRABASE Formula (France) . . . . . . . . . . . . . 2.1.5 Brilon-Wu Formula (Germany) . . . . . . . . . . . . . 2.1.6 HCM 2000 Formula (USA) . . . . . . . . . . . . . . . 2.2 Exit and Circle Capacities . . . . . . . . . . . . . . . . . . . . 2.3 Consideration of Pedestrian Crosswalks . . . . . . . . . . . . 2.3.1 Entry Capacities in Presence of Pedestrian Crosswalks 2.3.2 Exit Capacities in Presence of Pedestrian Crosswalks . 2.4 Some Concluding Remarks on Capacity Formulations . . . . . 2.5 Capacity Calculation at Saturation or Oversaturation Conditions of Entries . . . . . . . . . . . . . . . . . . . . . . 2.5.1 A Worked Example . . . . . . . . . . . . . . . . . . . 2.6 Calculation Procedures for Simple Capacity and Total Capacity . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 A Worked Example . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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15 15 17 19 22 27 32 34 36 36 37 44 44
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46 47
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52 54 56
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59 60 61 65
3 Waiting Phenomena at Steady State and Non-steady State Conditions . . . . . . . . . . . . . . . . . . . . 3.1 Some Results of Probabilistic Analysis of Queues 3.1.1 Some Remarkable Results . . . . . . . . 3.1.2 Concluding Remarks . . . . . . . . . . .
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xi
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Contents
3.2
Deterministic Analysis of Queues . . . . . . . . . . . . . . . . . 3.2.1 A Worked Example . . . . . . . . . . . . . . . . . . . . . 3.2.2 Some Remarkable Results . . . . . . . . . . . . . . . . . 3.2.3 Further Expressions of the Average Waiting Times . . . . . 3.2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 3.3 Time-Dependent Solutions and Waiting Phenomena . . . . . . . . 3.3.1 Time-Dependent Solutions for the Number of Users in the System and in the Queue. A Worked Example . . . . 3.3.2 Time-Dependent Solutions for the Average Time Spent in the System and in the Queue. A Worked Example 3.3.3 Generalized Time-Dependent Solutions . . . . . . . . . . 3.3.4 Time-Dependent Solutions for Traffic Peaks Between Two Periods at Steady-State Conditions . . . . . 3.3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67 70 74 86 87 88 91 94 97 98 104 105
4 Calculation of Waiting Times, Queue Lengths, and Levels of Service . . . . . . . . . . . . . . . . . . . . . 4.1 State Evolutions with Undersaturated Entries . . . . . 4.1.1 First Worked Example . . . . . . . . . . . . . 4.1.2 Second Worked Example . . . . . . . . . . . . 4.2 State Evolution with Saturated or Oversaturated Entries 4.2.1 First Worked Example . . . . . . . . . . . . . 4.2.2 Second Worked Example . . . . . . . . . . . . 4.3 Evaluation of Levels of Service . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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107 107 111 116 121 123 132 134 137
5 Evaluation of Roundabout Reliability . . . . . . . . . 5.1 Reliability and Performance Functions . . . . . . . 5.2 General Calculation Procedure of Reliability . . . . 5.2.1 A Worked Example . . . . . . . . . . . . . 5.3 Approximated Calculation Procedure of Reliability 5.3.1 A Worked Example . . . . . . . . . . . . . 5.4 Some Remarks . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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139 140 143 145 151 153 154 156
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
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Chapter 1
Calculation of Roundabouts: Problem Definition
By analyzing roundabouts, we mean the determination of efficiency measures when these intersections are to be used by a known traffic demand. The indices that we generally take into account are: Capacity and other related indicators Queue lengths Waiting times and delays In traffic engineering, queue lengths, waiting times, and delay values are also indicated as measures of effectiveness (MOE). Capacity determinations for entries refer to capacity, capacity rate, and reserve capacity, which is expressed in both absolute terms and in percentages. Capacity evaluations for the whole roundabout refer to simple capacity, total capacity, the mean of reserve capacity, and the mean of capacity rate at entries. Queues result from the waiting phenomena that drivers may suffer at the entries, and we estimate the length of the queues, measured by the number of vehicles, in terms of means and percentiles. Waiting times are the result of queuing up. They increase the traveling time because of the intersections along the route. Waiting times may refer to single users, as value or expected value1 , or, with appropriate means, to the whole intersection. When we calculate capacity, queue lengths, and waiting times, we must specify, for the observation period chosen, steadiness and variability of traffic demand and the presence of one or more entries that are saturated or oversaturated. This involves the analysis of the intersection with and without statistical equilibrium and, according to the state of the entries, the use of probabilistic, deterministic, or time-dependent models.
1 Expected value generally means the average of a random variable. The terms mean, average, and mathematical expectation (for a random variable) are synonymous.
R. Mauro, Calculation of Roundabouts, DOI 10.1007/978-3-642-04551-6_1, C Springer-Verlag Berlin Heidelberg 2010
1
2
1
Calculation of Roundabouts: Problem Definition
In the remaining part of this chapter, we will specify the assumptions and the terms for calculating roundabouts, and we will also give specific definitions of the performance indices that are briefly listed above. In the following chapters some of the procedures used to calculate roundabouts will be illustrated.
1.1 Statistical Equilibrium and Steady-State Conditions The operational conditions in the time of a roundabout – as is the case for any uncertain, real system – may be considered as a succession of states with probabilistic characterization. The description of this evolution requires, in its most complete forms, knowledge of the probability associated with each state of the system. This probability, for the same state may vary any time. In this case, we say that the system exists in a transient condition. If the probabilities of the states remain constant with time, we may say that the system has reached a statistical equilibrium, and we denote it to be in a steady-state condition. It is now worth noting that, to evaluate if a system is at the steady-state condition, we often choose not to assess the time-invariant state probability distribution. Thus, the assessment of a steady-state condition is only about the constancy with time of appropriate statistical values (e.g., means, variances, joint moments, and rorder moments, etc.) associated with one or more variables that evolve randomly and that are believed to be linked to the operational conditions of the whole.2 As we will see more clearly in the following chapters, on the basis of the abovementioned second procedure used to confirm a steady-state condition, a roundabout is deemed to be at a steady-state condition when entering traffic demand does not vary with time, and, additionally, when the traffic is systematically served by the roundabout without the occurrence of the incoming traffic congestion phenomenon i.e., average queue lengths and waiting times that are indefinitely increasing with time. We assume that all of the analyses performed below related to the operating conditions of roundabouts are based on the assumption that the circulatory roadway and the exits are always undersaturated. Therefore, the possible entry congestion must be related to one or more saturated or oversaturated entries.3 Now, if i = 1; 2; ...; n is the number of the intersection legs and t is time, traffic demand at the entries is generally expressed [2] by a vector (vector of demand volumes) 2 The two meanings correspond, in the random process theory, to steady-state conditions in restricted sense (strong steady-state conditions) and in broad sense (weak steady-state conditions), respectively. See, for example, [1]. 3 Evidently, it is assumed in this way that the absence of entry vehicular congestion is related to undersaturated entries.
1.1
Statistical Equilibrium and Steady-State Conditions
Qe (t) = [Qei (t)]
i = 1; 2; ...; n
3
(1.1)
and by a matrix of traffic percentages (percentage origin/destination matrix) PO/D (t) = [Pij (t)]
i, j = 1; 2; ...; n
(1.2)
The elements of the origin/destination matrix of traffic demand MO/D (t) from entries “i” to exits “j” of the intersection MO/D (t) = [Qij (t)]
i, j = 1; 2; ...; n
(1.3)
is obtained multiplying each element of the row “i” of matrix (1.2) by the corresponding element Qei (t) of vector (1.1). (See example in Sect. 1.1.1). That being stated, the condition for having a steady-state system, as previously formulated, equals to the set of the following relationships Qe (t + t) = Qe (t)
(1.4)
PO/D (t + t) = PO/D (t)
(1.5)
for any t and Qei (t) < Ci (t) i = 1; 2; ...; n (1.6) for any “t”. In Eq. (1.6) (entry undersaturation condition), Ci (t) represents entry capacities, as defined in the following Sect. 1.2. If Eqs. (1.4) and (1.5) are valid, we evidently also have MO/D (t + t) = MO/D (t)
(1.7)
for any t. In current technical practice, a steady-state condition is considered to be achieved, with undersaturated entries, if traffic demand at the intersection is constant for a finite period of time T, but long enough4 to allow the stabilization of the operative conditions of the roundabout around the constant mean values E [·] of state variables. In addition, the punctual values of state variables must be little dispersed around the mean values E [·]. It is worth recalling that we have used here the queue lengths and waiting times defined in the following paragraph Sect. 1.3 as state variables. When Eqs. (1.4) and (1.5) are valid, and considering the preservation of flows at the intersection, for the vector of the traffic flow in the circulatory roadway, 4 For the duration of T (in s) we generally use the following indication (Morse), T > √ max{1/ ( Ci /3600 − Qei /3600)2 } , with Qei and Ci expressed in hourly volumes.
4
1
Qc (t) = [Qci (t)]
Calculation of Roundabouts: Problem Definition
i = 1; 2; ...; n
(1.8)
and for the vector of exiting traffic flows at legs
we have
Qu (t) = Qui (t)] i = 1; 2; ...; n
(1.9)
Qc (t + t) = Qc (t)
(1.10)
Qu (t + t) = Qu (t)
(1.11)
for any t. In the worked example shown in Sect. 1.1.1, we explain how to deduce the vectors Qc (t) and Qu (t) starting with traffic demand at the intersection { Qe (t); PO/D (t)}. In technical practice, as is well known, the geometrical and functional design of a roundabout (or of any other infrastructural element) is generally conducted by assuming that traffic demand is constant with time and with reference to the traffic volume (peak-hour volume, PHV) related to a suitably chosen hour (such as, for example, between the 30th and the 100th peak hour of the year) [2]. For a generic entry, to obtain the design traffic volume, the peak-hour volume is divided by the peak hour factor, phf, to take into account traffic variations within the peak hour. Working in this way, the design traffic volume is equal to the equivalent hourly traffic volume of the peak subhourly rate of flow [2]. However, strictly speaking, this procedure is not correct. In fact, using the design traffic volume (obtained from the peak-hour traffic demand) which is considered to be applied indefinitely (in accordance with steady-state conditions), we have queue lengths, waiting times, and delays that are considerably greater than the ones actually applied. In addition, the values of these parameters tend to become infinite under critical conditions, that is when demand nears, equals, or exceeds the entry capacity (saturated or oversaturated entry). These circumstances cause an overdimensioning of the geometrical elements of the roundabout and an unrealistic evaluation of the Level of Service. In reality, traffic peaks and critical conditions occur for more or less long time intervals, so that they are consequently finished with limited effects. It is then suitable to base the evaluation on the demand flow trend inside the time period considered and, in addition, on the effective duration of traffic peaks. Thus, as we will see in Chap. 3, the transient conditions of the system must be thoroughly analyzed.
1.1.1 A Worked Example on Traffic Demand at a Roundabout Consider the roundabout in Fig. 1.1 and a steady-state condition in which traffic demand at the intersection, indicated by Eqs. (1.1) and (1.2), is the following
1.1
Statistical Equilibrium and Steady-State Conditions
5
Fig. 1.1 Traffic demand at roundabout (volumes in veh/h)
(volumes are expressed in veh/h) Qe1 Qe2 Qe3 Qe4 ≡ 350 400 300 ⎤ ⎡ ⎡ 0 P11 P12 P13 P14 ⎢ P21 P22 P23 P24 ⎥ ⎢ 0.25 ⎥ ⎢ ≡⎢ ⎣ P31 P32 P33 P34 ⎦ ≡ ⎣ 0.28 P41 P42 P43 P44 0.40
Qe ≡ PO/D
450
0.32 0 0.34 0.32
0.41 0.31 0 0.28
⎤ 0.27 0.44 ⎥ ⎥ 0.38 ⎦ 0
Multiplying each element in row “i” of the matrix PO/D by the corresponding Qei of vector Qe, we have the origin/destination matrix MO/D for the roundabout. (The terms are approximated to integer values.) ⎡
MO/D
Q11 ⎢ Q21 ≡⎢ ⎣ Q31 Q41
Q12 Q22 Q32 Q42
Q13 Q23 Q33 Q43
⎤ ⎡ 0 Q14 ⎢ 100 Q24 ⎥ ⎥≡⎢ Q34 ⎦ ⎣ 84 Q44 180
112 0 102 144
144 124 0 126
⎤ 94 176 ⎥ ⎥ 114 ⎦ 0
6
1
Calculation of Roundabouts: Problem Definition
Fig. 1.2 Traffic flows at the roundabout for the entry “i”
From matrix MO/D , due to traffic preservation on the circulatory roadway, it is possible to deduce (See Fig. 1.2) the volumes Qci in front of the entries and the volumes Qui exiting the roundabout with the following straightforward relationships: Qc1 = Q42 + Q43 + Q32 = 144 + 126 + 102 = 372 veh/h Qc2 = Q13 + Q14 + Q43 = 144 + 94 + 126 = 364 veh/h Qc3 = Q24 + Q21 + Q14 = 176 + 100 + 94 = 370 veh/h Qc4 = Q31 + Q32 + Q21 = 84 + 102 + 100 = 286 veh/h Qu1 = Q21 + Q31 + Q41 = 100 + 84 + 180 = 364 veh/h Qu2 = Q12 + Q32 + Q42 = 112 + 102 + 144 = 358 veh/h Qu3 = Q13 + Q23 + Q43 = 144 + 124 + 126 = 394 veh/h Qu4 = Q14 + Q24 + Q34 = 94 + 176 + 114 = 384 veh/h In the end, vectors (1.8) and (1.9) are, respectively: Qc ≡ 372 364 370 286 Qu ≡ 364 358 394 384 Since the example is conducted assuming the system to be in a steady-state condition, and, therefore, traffic demand is constant with time and the entries are undersaturated, vectors indicating the different ways to express demand coincide with the flow vectors that are actually traveling through the intersection: in this case, therefore, Qe indicates the traffic entering the roundabout, Qc indicates circulating traffic in front of the entries, and Qu indicates the exiting traffic. However, from now on, corresponding volumes of demand and the flows in transit will be indicated with the same symbol; the meaning of the symbol, when it is not explicitly indicated, can be easily inferred from the context.
1.2
Capacity and Capacity Indices
7
Evidently, calculation of the flows Qci and Qui may be conducted without the determination of matrix MO/D by using the elements of vector Qe and matrix PO/D directly. Therefore, due to the preservation of flows in the circulatory roadway and assuming, for the sake of simplicity, the flows that have the same leg as origin and destination to be null since they are generally very small, we have the following relationships: Qc1 Qc2 Qc3 Qc4
= Q42 + Q43 + Q32 = (P42 + P43 ) Qe4 + P32 Qe3 = Q13 + Q14 + Q43 = (P13 + P14 ) Qe1 + P43 Qe4 = Q24 + Q21 + Q14 = (P24 + P21 ) Qe2 + P14 Qe1 = Q31 + Q32 + Q21 = (P31 + P32 ) Qe3 + P21 Qe2
Qu1 = Q21 + Q31 + Q41 Qu2 = Q12 + Q32 + Q42 Qu3 = Q13 + Q23 + Q43 Qu4 = Q14 + Q24 + Q34
= P21 Qe2 + P31 Qe3 + P41 Qe4 = P12 Qe1 + P32 Qe3 + P42 Qe4 = P13 Qe1 + P23 Qe2 + P43 Qe4 = P14 Qe1 + P24 Qe2 + P34 Qe3
(1.12)
(1.13)
Therefore, it is easy to determine, starting with traffic demand, the circulatory roadway volumes and those traveling towards the exits for a roundabout with a number of legs i = 1, 2,. . ., any n. Values of the circulating flows Qci in front of each entry i, i ∈[1,. . ., n] (numbered in an anti-clockwise manner) are Qc,1 = Qe,n (Pn,2 + ... + Pn,n−1 ) + Qe,n−1 (Pn−1,2 + ... + Pn−1,n−2 ) + ... + Qe,3 P3,2 ...... Qc,n = Qe,n−1 (Pn−1,1 + ... + Pn−1,n−2 ) + Qe,n−2 (Pn−2,2 + ... + Pn−2,n−3 ) + ... + Qe,2 P2,1 (1.14) Qu,1 = Qe,2 P2,1 + ... + Qe,n Pn,1 ...... Qu,n = Qe,1 P1,n + ... + Qe,n - 1 Pn - 1,n
(1.15)
1.2 Capacity and Capacity Indices Capacity C of an entry is defined as the smallest value of the leg flow that causes the permanent presence of vehicles queuing up to enter. To calculate C, the roundabout is considered as a series, along the development of the junction, of T intersections, with yielding to the circulating flows with only one interaction i.e., the mutual contribution to the formation of circulating flows that affect each entry as conflicting flows. For a roundabout entry, capacity C may be expressed in the most general way as ˜ ˜ d ;˜τ;S) C = C(G;Q
(1.16)
8
1
Calculation of Roundabouts: Problem Definition
where ˜ is a set of variables representing the geometrical layout of the roundabout G (e.g., entry width and central island radius) or its configuration (e.g., number of circulatory roadway lanes and number of entry lanes); Qd is the traffic flow that disturbs the traffic entering the roundabout; Qd is the function of the entry flows (1.1) in general, by means of the circulating flows Qc (1.8) and the exiting flows Qu (1.9) (See Fig. 1.1); τ˜ is a set of psycho-technical times, relative to user behavior i.e., generally, critical gap Tc and follow-up time Tf 5 ; S˜ is a set of numerical constants that result from the calibration process of the capacity formula. A capacity formula is generally obtained in two ways [4]. a) It may be obtained by the calibration with empirical data of queuing theory models, i.e. models based on the gap-acceptance theory. b) It may be obtained with empirical regression techniques applied to sampling traffic data without using the queuing theory. However, specifying Eq. (1.16), today’s available6 capacity formulas may be classified into the following three types: a) the roundabout is characterized only by its configuration, represented by the number of circle lanes and leg lanes; b) the geometric design is taken into account at a reasonable level of detail; c) user behavior with critical gap Tc and follow-up time Tf is taken into account, along with geometric aspects. Examples of these three types of capacity formulas are dealt with in Chap. 2. According to some capacity formulas, the disturbing flows Qd are equal to circulating flows Qc ; according to other formulas, they are equal to appropriate linear combinations of the flows Qc and Qu . Starting with capacity Ci of an entry “i”, we will now define further capacity indices that are frequently used to characterize roundabout operational conditions: – Reserve Capacity (RC)i , which is equal to the difference between capacity Ci and traffic demand Qei at an entry 5T
is also called move-up time. For the definition of Tc and Tf , see, for example [3]. the international literature, as far as we know, 32 formulations of capacity have been found, eight of which are German (1992–2007), four are Swiss (1989–2006), two are American (1997– 2000), one is English (1980); three are French (1988–1997), three are Dutch (1992–1999); one is Polish (1996), one is Swedish (1996), one is Norwegian (1985), one is Finnish (2004), one is Danish (1999), one is Israeli (1997), two are Australian (1989–1998), two are Austrian (1997– 2001), and one is Portuguese (1996). f
6 In
1.3
Delays and Queue Lengths
9
(RC)i = Ci − Qei
(1.17)
– Percentage Reserve Capacity (RC%)i , given by (RC%)i =
(RC)i Ci − Qei · 100 = · 100 Ci Ci
(1.18)
– Percentage Capacity Rate (CR%)i , equal to 100 times the traffic intensity ρi = Qei /Ci
(CR%)i =
Qei · 100 Ci
(1.19)
We will now give further definitions of capacity related to the entire roundabout: 1) Simple Capacity (SC): with reference to a given traffic demand at the intersection, we define simple capacity SC as the first capacity value that is recorded at an entry for a uniform increase in the flows that make up the demand. In other words, when we calculate simple capacity, we also determine for a given vector (1.1) and matrix (1.2) the value of the maximum entering flow on each leg when one of the entry, in the uniform evolution of the system, becomes saturated. 2) Total Capacity (TC): we define total capacity TC with respect to a given percentage distribution of entering traffic (See matrix (1.2)) the sum of the flows from each entry (Qei ) that are simultaneously equal to capacity (Qei = Ci ). Therefore, Total Capacity indicates, for a given distribution of demand at the intersection (i.e., for a given determination of (1.2)), a concise measurement of the roundabout limited ability to serve traffic when each leg is saturated. The procedures to determine SC and TC will be illustrated in Chap. 2. Finally, capacity indices (absolute and in percentage) for the entire roundabout may be defined as the weighted means on the flows of the determinations obtained, respectively, with Eqs. (1.17), (1.18), and (1.19) for each roundabout entry.
1.3 Delays and Queue Lengths Delays at intersections along the route contribute to lost time during the travel. Figure 1.3 shows the components that make up the delay due to the presence of a roundabout along the route. The simplified time-distance diagram shown in Fig. 1.4 may replace the timedistance diagram shown in Fig. 1.3 in order to evaluate delays. In Fig. 1.4, the horizontal line ws indicates, using the term of the queuing theory [5], the time spent in the system or the delay of the vehicle in the sense of traffic engineering.
10
1
Calculation of Roundabouts: Problem Definition
Fig. 1.3 Delays due to the presence of a roundabout along the route
Fig. 1.4 Simplified space versus time diagram
It results from the addition of two rates: • queue waiting time wc , i.e., the time that a single vehicle spends from the moment it joins the queue till it gets to the head of the queue (i.e., to the yielding line); • service time Ts , which is the time between the moment the vehicle reaches the head of the queue (i.e., the yielding line) and the moment it enters the circulatory roadway to perform the desired maneuver.
1.3
Delays and Queue Lengths
11
Finally, ws = wc + Ts
(1.20)
Now, with the model generally used to obtain the delay w (total delay) (See Fig. 1.4), the times Tdec and Tacc must be considered together with ws . They indicate the deceleration phase when approaching the intersection and the acceleration phase when crossing and exiting the intersection, respectively. In the roundabout of Fig. 1.4, Tdec = t2 – t1 , and Tacc = t4 – t3 . Thus, the delay at the intersection w, is function, among other things, of w = f(ws ;Tdec ;Tacc ;(.);(.)...)
(1.21)
In conclusion, total delay is the difference between the traveling time spent in presence of the intersection and the traveling time that would be spent in absence of the intersection. The difference wg wg = w − ws
(1.22)
is called geometric delay. The determination of w is used, for example: (a) to evaluate costs related to traffic assignment to road networks; (b) to make the comparative analysis used to change the layout of an intersection; (c) to compare roundabouts of different geometry. To evaluate the quality of circulation at given intersections, it is, instead, sufficient to calculate only the time spent in the system ws provided by Eq. (1.20). We will henceforth analyze only ws . 7 However, to calculate geometric delay, see, for example [6]. Ls at an entry indicates the number of waiting vehicles in the system, i.e., including the vehicle in service8 (See Fig. 1.5). The length of the queue Lc is made up, instead, of the number of vehicles waiting behind the vehicle in service, e.g., in the case of a roundabout with single-lane entries, Lc = Ls – 1. Ls is the number of users in the system. ws , wc , Ls, and Lc are random variables.
7 In technical practice, but sometimes also in scientific literature, “delay” does not mean only w given by Eq. (1.21), but also ws expressed by Eq. (1.20) that, as we have already pointed out, is part of w and is relative only to the time spent queuing up at the beginning of the maneuver to enter the circle. 8 The vehicle in service is the first vehicle in the queue, i.e., the nearest to the yielding line, waiting to enter the circle.
12
1
Calculation of Roundabouts: Problem Definition
Fig. 1.5 Number of vehicles in the system and number of vehicles in the queue
To determine the effectiveness measures of roundabouts, we need the mean values E[ws ] of ws and E[wc ] of wc and, for Ls and Lc , in addition to the means E[Ls ] and E[Lc ], we need suitable percentiles Ls,p and Lc,p . The order p of the percentiles may be established in relation to the particular problem under examination. As we have already pointed out in Sect. 1.1, the determinations of E[ws ], E[wc ], E[Ls ], E[Lc ], Ls,p , Lc,p require the characterization of the state of the system in terms of the presence or absence of steady-state conditions. In a steady-state condition, the mean (average) E[·] is a mathematical expectation of a random variable. To define, instead, the mean in the absence of steady-state conditions, it is necessary to take into account that, under these conditions, the queue mean length varies according to the change of the system conditions (traffic demand variation and/or entry states).
References
13
In particular, if the absence of a steady-state condition is connected to saturation or oversaturation at the entry of concern for a period T, at the beginning of T, queues shorter than those at the end of the same period will be recorded. For the time spent in the system and waiting time in the queue, it follows that the same behavior, i.e. a shorter duration at the beginning of T compared to that recorded at its end, may be recorded. Thus, for example, the average waiting time in the queue E[wc ] related to T is defined as E[wc ] =
wc (t0 ) + wc (t) 2
(1.23)
where wc (t0 ) and wc (t) are the waiting times in the queue of two vehicles joining the queue at the beginning t0 and at the end t = t0 + T of the observation period T. Equation (1.23) may be also considered as an estimate of the time that an user may spend, on average, in the queue when he or she arrives at half of the period of entry saturation or oversaturation. In addition, it is important for practical applications to evaluate the percentile Ls,p of the number of users in the system at the end of T, for the situations connected to saturated or oversaturated entries. The estimation of a suitable Ls,p in an entry to a roundabout is an essential task for the following reasons: a) it is an indicator of traffic quality. b) it is useful to design and validate the geometric layout of the intersection. c) it helps to prevent traffic jams into upstream intersections. Chapter 3 will be dedicated to the criteria to calculate the waiting phenomena parameters at roundabout entries.
References 1. Hsu, H. W., Probability, Random Variables and Random Processes, MacGraw-Hill, New York, 1997. 2. Roess, P. R., McShane, R. W., Prassas, S. E., Traffic Engineering – 2nd Ed., Prentice Hall, Upper Saddle River, New Jersey, 1998. 3. Louah, G., Priority Intersections: Modelling, in “Concise Encyclopedia of Traffic and Transportation System” (Papageorgiu, M., Editor), Pergamon Press, Oxford, 1991. 4. Brilon, W. (Editor), Intersections without Traffic Signals – Vol. I and II, Springer-Verlag, Heidelberg, 1998 and 1991. 5. Kleinlock, L., Queuing Systems – Vol. I, John Wiley and Sons, New York, 1975. 6. Teakratok, T., Modern Roundabout for Oregon, Oregon Department of Research Unit, 1998.
Chapter 2
Capacity Evaluation
In the previous Sect. 1.2, the definitions of capacity C of an entry, simple capacity SC, and whole or total capacity TC were discussed. The section also contains further definitions regarding reserve capacity indices (absolute RC, percentage RC%, and percentage Capacity Rate CR%). As we will see in the following Sect. 2.1, the determination of Capacity C i at the entries “i” is easy if the entries are all at undersaturation conditions (RC)i = Ci − Qei > 0, i.e., ρi = Qei Ci < 1 ∀i . On the other hand, if one or more of the entries “i” are not undersaturated, the determination of their capacities Ci is not straightforward. The determinations of simple capacity SC and total capacity TC are not straightforward. In all these cases, it is necessary to use specific, iterative calculation methods, which will be dealt with in the following Sects. 2.5 and 2.6.
2.1 Capacity Calculation at Steady-State Conditions As already described, by specifying Eq. (1.16) in Chap. 1, the capacity formulas that are currently available may be classified into three types: a) the roundabout is characterized only by its configuration, represented by the number of circle lanes and leg lanes; b) the roundabout geometry is taken into account in somewhat detailed way; c) we take into account, together with geometric aspects, the users’ behaviors thanks to psycho-technical times Tc , critical gap, and follow-up time Tf . To provide some examples of the type of relationships of the above-mentioned classifications (a; b; c), six capacity formulas are now presented and implemented. Two formulas of type (a) that were developed by Brilon et al. (Germany) and by Bovy et al. (Switzerland) are presented in which the roundabout configuration is therefore represented by the number of lanes at the entries and in the circle. In addition, according to Brilon’s formula, the disturbing flow Qd is represented only R. Mauro, Calculation of Roundabouts, DOI 10.1007/978-3-642-04551-6_2, C Springer-Verlag Berlin Heidelberg 2010
15
16
2 Capacity Evaluation
by the circulating flow Qc in front of the entries, whereas, according to Bovy’s formula, Qd is a linear combination of Qc and exiting traffic Qu .1 The third formula presented, developed by TRRL (United Kingdom), belongs to set (b) and requires, with respect to the two previous cases, a more detailed characterization of the roundabout geometry and uses the size of the main planimetric elements of the roundabout as input; in this formula, the disturbing flow is represented only by the circulating flow Qc . The fourth formula presented, the GIRABASE procedure (France), belongs to set (c). In fact, it is necessary to determine some geometric values of the roundabout and to use a pre-fixed follow-up time value Tf to implement the formula. In this case, the disturbing flow is represented by a linear combination of the circulating flow with the exiting flow. The fifth formula, developed by Brilon and Wu (Germany), belongs to set (c), and it considers capacity as a function of the roundabout configuration, rendered in terms of number of lanes at the entries and in the circle and as function of the users’ psycho-technical attitudes, expressed by determining the critical gap Tc and follow-up time Tf 2 values. In this case, Qd is set equal to Qc . The sixth formula also belongs to set (c). It is included in the Highway Capacity Manual HCM 2002 [1]. The capacity formulas available in the literature generally express entering flows in passenger car unities (pcu/h), and, using the same measure, entry capacities are obtained. To express flows Qe in pcu/h, vehicles other than passenger cars are generally treated as follows: 1 truck, bus = 1.5 pcu; 1 truck + trailer = 2.0 pcu; 1 motorcycle = 0.5 pcu; 1 bicycle = 0.5 pcu. When formulas make use of the number of lanes, one should reason as follows. According to regulations that are largely accepted, circle lanes must not have road markings; for this reason, the phrase “number of circle lanes” is meant to be the number of circulating vehicles rows that can be accommodated on the circulatory roadway. In all the following examples, we assume that all the intersections under examination are at steady-state conditions. We wish to recall (see Sect. 1.1) that a steady-state condition, as we mean it here, is achieved if entries are undersaturated and traffic demand at each leg remains constant for a time period T of a suitable size. In other words, T must be long enough to allow the operating conditions of the intersection to become steady with constant mean values of state variables. In addition, the punctual values of state variables must be little dispersed around the mean values.
1 2
For Qe , Qu , and Qc , see Fig. 1.2 in Chap. 1. To evaluate Tc and Tf , see, for example, [3].
2.1
Capacity Calculation at Steady-State Conditions
17
2.1.1 Brilon-Bondzio Formula (Germany) The capacity of an entry is represented by the simple linear relationship [2, 3] C = A − B · Qc pcu h
(2.1)
where A and B are obtained from Table 2.1, depending on the numbers of entry and circle lanes. Equation (2.1) is valid for roundabouts with external diameters Dext that range from 28 to 100 m. In Eq. (2.1), the disturbing traffic Qd coincides with the circulating flow Qc in front of the entry for which C is determined. Figure 2.1 shows the representation of Eq. (2.1) for all the geometric configurations for which it is valid. As an example, consider a four-legged roundabout with a double-lane circle and double-lane entries. Table 2.1 Parameters values for Brilon-Bondzio capacity formula Circle lane number
Entry lane number
A
B
Sample size
3 2 2–3 1
2 2 1 1
1409 1380 1250 1218
0.42 0.50 0.53 0.74
295 4574 879 1504
Fig. 2.1 Capacity C versus circulating flow Qc according to Brilon-Bondzio capacity formula
18
2 Capacity Evaluation
The entering flow vector (pcu/h) is Qe = [322, 252, 329, 408]. The origin/destination matrix relative to the time considered is the following (the traffic volumes in matrix MO/D are expressed in pcu/h): ⎡
MO/D
⎤ 0 82 116 124 ⎢ 74 0 92 86 ⎥ ⎥ ≡⎢ ⎣ 106 96 0 127 ⎦ 128 141 139 0
From the entering flows vector Qe , with Eqs. (1.12) and (1.13) in Chap. 1, we obtain the exiting flows (Qu ) and circulating flows in front of each entry (Qc ) for each leg, i.e., i = 1, 2, 3, 4, of the roundabout: Qu1 = 308 pcu/h Qu2 = 319 pcu/h Qu3 = 347 pcu/h Qu4 = 337 pcu/h
Qc1 = 376 pcu/h Qc2 = 379 pcu/h Qc3 = 284 pcu/h Qc4 = 276 pcu/h
On the basis of Eq. (2.1), with A = 1380 and B = 0.50 (See Table 2.1), the capacities for each entry are: C1 C2 C3 C4
= = = =
1380 − 0.5 · 1380 − 0.5 · 1380 − 0.5 · 1380 − 0.5 ·
Qc1 Qc2 Qc3 Qc4
= 1192 pcu/h = 1190 pcu/h = 1238 pcu/h = 1242 pcu/h
Then, we evaluate the capacity indices for each entry (See Sect. 1.2). The reserve capacities are: (RC)1 (RC)2 (RC)3 (RC)4
= = = =
C1 − Qe1 C2 − Qe2 C3 − Qe3 C4 − Qe4
= = = =
1192 − 322 = 870 pcu/h 1190 − 252 = 938 pcu/h 1238 − 329 = 909 pcu/h 1242 − 408 = 834 pcu/h
Percentage Capacity Rates (CR%) for each entry are (CR% )1 = (Qe1 /C1 ) · (CR% )2 = (Qe2 /C2 ) · (CR% )3 = (Qe3 /C3 ) · (CR% )4 = (Qe4 /C4 ) ·
100 = (322/1192) 100 = 27.0% 100 = (252/1190) 100 = 21.2% 100 = (329/1238) 100 = 26.6% 100 = (408/1242) 100 = 32.9%
2.1
Capacity Calculation at Steady-State Conditions
19
For the roundabout as a whole, the following mean values are obtained n
RC =
i=1
(RC)i · Qei n
= Qei
870 · 322 + 938 · 252 + 909 · 329 + 834 · 408 ∼ = 881 pcu/h 322 + 252 + 329 + 408
i=1 n
(CR% ) =
i=1
(CR% )i · Qei n
= Qei
i=1
27.0% · 322 + 21.2% · 252 + 26.6% · 329 + 32.9% · 408 ∼ = = 27.6% 322 + 252 + 329 + 408
2.1.2 Bovy et al. Formula (Switzerland) This formula is recommended for roundabouts (to be used in urban and suburban environments) with a non mountable central island, of small dimensions (maximum internal diameter Dint = 18 – 20 m) [3]. The circle external diameter Dext varies generally from 24 to 34 m, and there are flared entries, i.e., there are more lanes next to the stop line to make the choice of the desired direction easier. An entry capacity is determined with the relationship: C =
8 1 · (1500 − · Qd ) (pcu/h) γ 9
(2.2)
where γ is a parameter that allows taking into account the number of entry lanes, and its value is: – γ = 1 for one lane; – γ = 0.6–0.7 m for two lanes (according to smaller or larger entry dimensions), and it is generally set at 0.667; – γ = 0.5 for three lanes. Qd is the disturbing traffic determined as: Qd = α · Qu + β · Qc (pcu/h) where (See Fig. 2.2): Qu = exiting traffic; Qc = circulating traffic in front of the exit being considered.
(2.3)
20
2 Capacity Evaluation
Fig. 2.2 Distance between the exiting conflicting point (A) and entering point (B)
Coefficients α and β are related to the geometry of the roundabout and take into account the distance between the exiting conflicting points (A) and entering points (B) that are conventionally identifiable on the circle (Fig. 2.2) and the number of circular lanes, respectively. Due to the ability to simulate the roundabout, it was possible to establish that, in accordance with the intuitively obvious inverted proportionality relationship between and the percentage disturb due to the vehicles exiting the intersection, α decreases with until, for > 28 m, the exiting vehicles do not disturb the entering vehicles (α = 0). Figure 2.3 shows three behaviors of the value α as a function of the distance . The line “a” is relative to a circle flow speed of 20–25 km/h; lines “b” and “c” border the band above and below “a” when V > 20–25 km/h (greater disturb) and when V < 20–25 km/h (smaller disturb), respectively.
Fig. 2.3 Values of parameter α versus the distance between the exiting and entering conflicting points shown in Fig. 2.2
2.1
Capacity Calculation at Steady-State Conditions
21
For β, which takes into account the reduction effect that the presence of more than one circle lane has on the conflicting flow caused by the circulating traffic, the following values are provided: β = 0.9–1.0 for one lane; β = 0.6–0.8 for two lanes; and β = 0.5–0.6 for three lanes. To calculate the number of equivalent passenger car units (pcu), the following values are suggested: one bike or motorbike in the circle = 0.8 pcu one entering bike or motorbike = 0.2 pcu one heavy vehicle or bus = 2.0 pcu The Swiss Standards on roundabouts use the formula proposed by Bovy et al. and use the following two capacity indices as the roundabout efficiency indicators. The calculation of these indices and their meanings are straightforward, and they must always be determined together. The calculations are performed as shown below: – Capacity Rate used at entries (CRUe ): CRUe = (γ · Qe /Ce ) · 100 (%)
(2.4)
– Capacity Rate used at the conflicting point (CRUc ): CRUc = [(γ · Qe + 8/9 · Qd )/1500] · 100 (%)
(2.5)
Now, we present an example of implementation of the procedure for the roundabout examined in the previous Sect. 2.1.1 (a four-legged roundabout with a double-lane circle and double-lane entries). We assume that the distance between the conflicting points is equal to 20 m in front of each leg. The traffic data are the same as the traffic data used in the example in Sect. 2.1.1. The circle flow speed is assumed to be 20–25 km/h. The value of the parameter α is determined from the diagram shown in Fig. 2.3, along the line “a”, as a function of the distance = 20 m for all the legs. The result is α = 0.14. Since the circle has two lanes, β is equal to 0.7. Thus, disturbing flows in front of the legs can be determined in the following way: Qd1 Qd2 Qd3 Qd4
= = = =
α· α· α· α·
Qu1 + β · Qu2 + β · Qu3 + β · Qu4 + β ·
Qc1 Qc2 Qc3 Qc4
= 0.14 · = 0.14 · = 0.14 · = 0.14 ·
308 + 319 + 347 + 337 +
0.7 · 0.7 · 0.7 · 0.7 ·
376 379 284 276
∼ = ∼ = ∼ = ∼ =
307 pcu/h 311 pcu/h 249 pcu/h 242 pcu/h
22
2 Capacity Evaluation
Therefore, with Eq. (2.2), we can determine the capacity values at entries, using γ = 0.667 for all (because of double-lane entries), as shown below: C1 C2 C3 C4
= 1/γ · = 1/γ · = 1/γ · = 1/γ ·
(1500 − 8/9 · (1500 − 8/9 · (1500 − 8/9 · (1500 − 8/9 ·
Qd1 ) Qd2 ) Qd3 ) Qd4 )
= = = =
1/0.667 · 1/0.667 · 1/0.667 · 1/0.667 ·
(1500 − 8/9 · (1500 − 8/9 · (1500 − 8/9 · (1500 − 8/9 ·
307) 311) 249) 242)
∼ = ∼ = ∼ = ∼ =
1840 pcu/h 1834 pcu/h 1917 pcu/h 1926 pcu/h
Finally, the values CRUe and CRUc are: CRUe1 CRUe2 CRUe3 CRUe4 CRUc1 CRUc2 CRUc3 CRUc4
= = = = = = = = = = = =
(γ · Qe1 /Ce1 ) · 100 = (0.667 · 322/1840) 100 = (γ · Qe2 /Ce2 ) · 100 = (0.667 · 252/1834) 100 = (γ · Qe3 /Ce3 ) · 100 = (0.667 · 329/1914) 100 = (γ · Qe4 /Ce4 ) · 100 = (0.667 · 408/1926) 100 = (γ. Qe1 + 8/9 · Qd1 )/1500 · 100 (0.667 · 322 + 8/9 · 307)/1500 · 100 = 32.5% (γ. Qe2 + 8/9 · Qd2 )/1500 · 100 (0.667 · 252 + 8/9 · 311)/1500 · 100 = 29.6% (γ. Qe3 + 8/9 · Qd3 )/1500 · 100 (0.667 · 329 + 8/9 · 249)/1500 · 100 = 29.4% (γ. Qe4 + 8/9 · Qd4 )/1500 · 100 (0.667 · 408 + 8/9 · 242)/1500 · 100 = 32.5%
11.7% 9.2% 11.4% 14.1%
2.1.3 TRRL Formula (United Kingdom) With the TRRL formula, capacity C of a generic entry is determined as a function of the leg and circle geometric parameters and of the circulating flow in the circle (Qc ) in front of the entry [4]. The relationship was developed by Kimber, and it is based on experimental observations of a large number of operating roundabouts in England. It has the following linear form: C = k · (F − fc Qc ) (pcu/h) where: F = 303 · x2 fc = 0.210 · tD · (1 + 0.2 · x2 ) k = 1 – 0.00347 · (–30) – 0.978 · (1/r–0.05) 1 2 · [1 + exp((D − 60)/10)] (e − v) x2 = v + (1 + 2 · S) tD = 1 +
S = 1.6 (e–v)/ = (e–v)/
(2.6)
2.1
Capacity Calculation at Steady-State Conditions
23
Table 2.2 Geometric parameters used by the TRRL formula Parameter
Description
Range values
e v e v u , S r D = Dext W L
Entry width Lane width Previous entry width Previous lane width Circle width Flare mean length Sharpness of the flare Entry bend radius Entry angle Inscribed circle diameter Exchange section width Exchange section length
3.6–16.5 m 1.9–12.5 m 3.6–15.0 m 2.9–12.5 m 4.9–22.7 m 1–∞ m 0–2–9 3.4–∞ m 0–77◦ 13.5–171.6 m 7.0–26.0 m 9.0–86.0 m
Table 2.2 shows the geometric parameters, the respective symbols used in the procedure, and their range [4]. The main indications contained in [4] for the determination of such parameters are now presented. However, this determination can sometimes be rather difficult because of the particular geometric configurations of the roundabout. To illustrate the geometric elements that are used in the formula, it is useful to observe Figs. 2.4, 2.5, and 2.6. They are taken from the original work [4], and they show the left-side driving in the United Kingdom requiring that travel in the circle be clockwise. To apply the procedure to counterclockwise roundabouts that are used for right-side traffic, homologous symmetrical elements must be used. The width of the entry (e) is determined along the perpendicular line traced from point A to the external edge (See Fig. 2.4). The width of the entry lane (v) must be determined upstream of the leg widening next to the entry along the perpendicular line traced from the axis of the roadway to the external edge. The width of the circulatory roadway (u) represents the distance between the splitter island at legs (point A) and the central island. The entry radius (r) is the smallest bend radius of the external edge next to the entry. The width of the weaving section (W) is the shortest distance between the central island and the external edge in the stretch between an entry and the following exit. The weaving section (L) is defined as the shortest distance between the splitter islands at the legs of two successive entries. The mean length of the flare can be determined using either of the two parameters or . Figure 2.5 shows the geometric constructions for their determination. In both cases, by tracing a line parallel to the curve HA (at a distance v from it), we can determine the curve GD, which intersects the segment AB (which represents the entry width) at point D; the length corresponds to the segment CF, determined along the perpendicular line that passes through C (mean point of segment BD) of segment AB (with F as intersection point between the above-mentioned perpendicular and the curve GD); length corresponds to segment CF along a curve parallel to
24
Fig. 2.4 Geometric elements used in the TRRL formula [3]
Fig. 2.5 Geometric construction for the determination of and [3]
2 Capacity Evaluation
2.1
Capacity Calculation at Steady-State Conditions
25
Fig. 2.6 Geometric construction for the determination of the entry angle [4]
the external edge BG and passing through C (with F as intersection point between the above-mentioned curve and the curve GD). Between and , the approximate relationship = 1.6 is valid within the allowed variability for the practical use of geometric parameters. The entry angle (), which represents the conflicting angle between the entering flows and the circulating flows, must be determined according to the straightforward indications shown in Fig. 2.6. Now, we present an example of the application of the procedure. Consider a four-legged roundabout with the following features: – inscribed circle diameter: D = 45 m; – entry width: e = 4.8 m; – lane width: v = 3.5 m; – flare mean length = 25 m; – entry radius r = 40 m; – entry angle: = 60◦ . Traffic data (expressed in pcu/h) are summarized in the matrix O/D: ⎡
MO/D
⎤ 0 150 300 200 ⎢ 200 0 150 350 ⎥ ⎥ ≡⎢ ⎣ 350 150 0 150 ⎦ 300 250 200 0
26
2 Capacity Evaluation
From matrix O/D, the circulating flows in front of each entry can be obtained (Eq. (2.12)): Qc1 Qc2 Qc3 Qc4
= = = =
600 pcu/h 700 pcu/h 750 pcu/h 700 pcu/h
Then, we can determine the calculation parameters of each leg using Eq. (2.6) since all of the geometric parameters relative to each entry are the same: S = 1.6 · (e − v)/ = 1.6 · (4.8 − 3.5)/25 = 0.083 x2 = v + (e − v)/(1 + 2 · S) = 3.5 + (4.8 − 3.5)/(1 + 2 · 0.083) = 4.615 F = 303 · x2 = 303 · 4.615 = 1398 tD = 1 + 0.5/[1 + exp ((D − 60)/10)] = 1 + 0.5/[1 + exp((45 − 60)/10)] = 1.409 fc = 0.210 · tD · (1 + 0.2 · x2 ) = 0.210 · 1.409 · (1 + 0.2 · 4.615) = 0.569 k = 1 − 0.00347 · ( − 30) − 0.0978 · (1/r − 0.05) = = 1 − 0.00347.(60 − 30) − 0.978 · (1/40 − 0.05) = 0.920 Therefore the capacity formula for all the legs can be written as: C = k · (F − fc Qc ) = 0.920 · (1398 − 0.569 · Qc ) = 1286 − 0.523 · Qc The capacity values determined for the four entries are: C1 C2 C3 C4
= = = =
972 pcu/h 919 pcu/h 893 pcu/h 919 pcu/h
Among the further applications of the TRRL capacity formula, we illustrate the determination of a roundabout entry width. The entering flow of an entry is set to Qei = 800 pcu/h; the circulating flow in front of the entry is set to Qci = 1100 pcu/h. The geometric data relative to the entry are the following: – inscribed circle diameter D = 40 m; – entry lane width v = 7.3 m; – flare mean length = 20 m; – entry bend radius r = 25 m; and – entry angle = 30◦ . We want to determine the entry width “e” necessary, for example, to ensure a reserve capacity equal to the entering flow.
2.1
Capacity Calculation at Steady-State Conditions
27
Therefore, we have: k = 1 − 0.00347 · (30 − 30) − 0.978 · (1/25 − 0.05) = 1.01 tD = 1 + 0.5/[1 + exp((40 − 60)/10)] = 1.44 fc = 0.210 · tD · (1 + 0.2 · x2 ) = 0.210 · 1.44 · (1 + 0.2 · x2 ) F = 303 · x2 The entry capacity necessary to have a reserve capacity equal to the value of the entering flow is equal to 2 · 800 = 1600 pcu/h. Therefore, we have: C = k · (F − fc Qc ) 1600 = 1.01 · [303 · x2 − 0.210 · 1.44 · (1 + 0.2 · x2 ) · 1100] 1600 = 306.03 · x2 − 335.97 − 67.19 · x2 1935.97 = 238.84 · x2 x2 = 8.11 On the other hand, we can also determine: S = 1.6(e − v)/ = 1.6(e − 7.3)/20 (e − v) (e − 7.3) x2 = v + = 7.3 + (1 + 2 · S) [1 + 2 · 1.6 · (e − 7.3)/20] Equating the two expressions for x2 , we obtain the value of the width requested, which is e = 8.23 m.
2.1.4 GIRABASE Formula (France) GIRABASE is the commercial software currently used in France to determine the capacity of a roundabout. It was developed by CETE de l’Ouest of Nantes and was accepted by CERTU and by SETRA3 [5]. The final version of the formula was written after it was tested by Urbahn in Germany in 1996 and updated by Guichet in 1997. It was developed by treating traffic data collected by observing the entries of roundabouts (operating at saturation conditions) with statistical regression techniques. In particular, GIRABASE software’s empirical regression equations are based on the counting of 63000 vehicles during 507 saturated operation periods of 5–10 min in 45 different roundabouts [6]. The procedures can be used for all types of roundabouts (from small roundabouts to large roundabouts) located in urban or rural areas, with the number of legs ranging from three to eight and with one, two, or three circle lanes and entry lanes [7].
3 CETE: Centre d’Etudes Techniques de l’Equipement; CERTU: Centre d’Etudes sur les Réseaux, les Transport, l’Urbanisme et les constructions publiques; SETRA: Service d’Etudes Techniques des Routes et Autoroutes.
28
2 Capacity Evaluation
Fig. 2.7 Traffic flows and geometric elements for the GIRABASE formula
Figure 2.7 shows the traffic flows and the geometric elements of the roundabout considered in the procedure; Table 2.3 shows the ranges of the geometric elements for the application of the procedure. The formula for the determination of entry capacity (pcu/h), based on the exponential regression technique, is the following: C = A · e−CB ·Qd
(2.7)
with 3600 A= Tf
Le 3.5
0.8 (2.8)
Tf = follow-up time = 2.05 s; Le = width of the entry in proximity to the roundabout, determined perpendicularly to the entry direction (m); CB = coefficient that is 3.525 for urban areas and 3.625 for rural areas;
Table 2.3 Range of the roundabout geometric elements for the application of the GIRABASE procedure Parameter
Description
Range values
Le Li Lu LA Ri
Entry width Splitter island width Exit width Circle width Central island radius
3–11 m 0–70 m 3.5–10.5 m 4.5–17.5 m 3.5–87.5 m
2.1
Capacity Calculation at Steady-State Conditions
Qd = Qu · ka · 1 −
Qu Qc + Qu
29
+ Qci · kti + Qce · kte
(2.9)
Qd = disturbing flow in front of the entry (pcu/h); Qu = exiting flow (pcu/h); Qc = Qci + Qce = circulating flow in front of the entry (pcu/h); Qci = traffic rate Qc on the inner circle lane (pcu/h); Qce = traffic rate Qc on the outer circle lane (close to the entry) (pcu/h);
ka =
⎧ R ⎨ Ri + iLA − ⎩
Li Limax
0
per Li < Li max in the other cases
Ri = central island radius (m); LA = circle width (m); Li = splitter island width at legs (m); Limax = 4,55 ·
kti = min
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Ri +
LA 2
160 LA · (Ri + LA) 1
⎧ 2 (LA − 8) Ri ⎪ ⎪ ⎨1 − · LA (Ri + LA) kte = min ⎪ ⎪ ⎩ 1
As an example of the application of the procedure, consider a four-legged roundabout with a double-lane circle and single-lane entries, located in a rural environment. It has the following geometric features: – external diameter: Dext = 50 m; – entry width: Le = 4.0 m; – splitter island width at legs: Li = 7.0 m; – circle width: LA = 10.0 m. The value Ri of the central island outer radius is determined with the relationship:
Ri =
Dext − 2 · LA = 15 m 2
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2 Capacity Evaluation
Since the roundabout is located in a rural area, the coefficient CB is 3.625. The value of parameter A, to be determined by means of Eq. (2.8), is the same for all entries and is equal to: A=
3600 Tf
Le 3.5
0.8 =
3600 2.05
4.0 3.5
0.8 = 1954
The traffic data from the TRRL procedure example (Sect. 2.1.3) are used here, and they are represented by the following matrix O/D: ⎡
MO/D
⎤ 0 150 300 200 ⎢ 200 0 150 350 ⎥ ⎥ ≡⎢ ⎣ 350 150 0 150 ⎦ 300 250 200 0
MO/D can be used to determine the values of the flows exiting from each leg and circulating in the circle in front of each entry: Qu1 Qu2 Qu3 Qu4
= 850 pcu/h = 550 pcu/h = 650 pcu/h = 700 pcu/h
Qc1 Qc2 Qc3 Qc4
= 600 pcu/h = 700 pcu/h = 750 pcu/h = 700 pcu/h
Regarding the traffic rates Qci and Qce , we assume that about 70% of the circulating flow travels on the outer circle lane and 30% travels on the inner circle lane in front of each entry. Thus, we have the following values: Qci1 Qci2 Qci3 Qci4
= = = =
420 pcu/h 490 pcu/h 525 pcu/h 490 pcu/h
Qce1 Qce2 Qce3 Qce4
= = = =
180 pcu/h 210 pcu/h 225 pcu/h 210 pcu/h
For kti and kte we calculate: 160 ;1 = min {0.57; 1} = 0.57 kti = min LA · (Ri + LA) 2 (LA − 8) Ri ;1 = min {0.92;1} = 0.92 kte = min 1 − · LA (Ri + LA) To determine the disturbing flows in front of each leg, we must also determine the coefficient ka that is the same for all the legs as function of Limax Limax = 4.55 · ka =
Ri +
LA 2
= 4.55 ·
15 +
10 2
= 20.35 m > Li = 4.0 m
Ri 15 Li 4.0 = − − = 0.256 Ri + LA Limax 15 + 10 20.35
2.1
Capacity Calculation at Steady-State Conditions
31
Therefore, with Eq. (2.9) we determine the disturbing flows in front of each leg. They are Qd1 = Qu1 · ka · 1 −
Qu1 Q c1 +Qu1
+ Qci1 · kti + Qce1 · kte = 850 = 850 · 0.256 · 1 − + 420 · 0.57 + 180 · 0.92 420 + 850 = 526 pcu/h
and, similarly, Qd2 = 587 pcu/h Qd3 = 634 pcu/h Qd4 = 598 pcu/h. Finally, from Eq. (2.7), the entry capacity values C1 = A · e−CB ·Qd1 = 1954 · e−3.625·526 = 1151pcu/h and, similarly, C2 = 1082 pcu/h C3 = 1032 pcu/h C4 = 1070 pcu/h Instead of applying the GIRABASE procedure, a simplified capacity formula is used (CERTU [8]) for urban French roundabouts. It is considered suitable for medium-sized and large-sized roundabouts (with a central island diameter from 20 to 60 m) with single-lane entries, symmetrical location of the legs, and a balanced entering traffic demand. In these cases, the entry capacity can be determined with the relationship: 5 · Qd 6
(2.10)
Qd = a · Qc + b · Qu
(2.11)
C = 1500 − as function of the disturbing flow
with a. variable, as function of the central island radius, between 0.9 and 0.7 for Ri < 15 m and > 30 m, respectively. b. variable, as function of the splitter island width at legs, between 0 and 0.3 for Li > 15 m and Li = 0 m, respectively.
32
2 Capacity Evaluation
Finally, it is worthwhile to note that, in France, the GIRABASE procedure long ago replaced the other formulas developed by various French road research centers. Therefore, even the SETRA formula (See Sect. 5.2.1), which is still used extensively by Italian engineers, is no longer used in France.
2.1.5 Brilon-Wu Formula (Germany) In Germany, based on an idea from Tanner, in 1997, Brilon and Wu proposed the following formula for the calculation of capacity C (pcu/h) of a roundabout entry (required by German Standard HBS 2001) [9]:
· Qc /3600 C = 3600 · 1 − nc
nc
ne Tf · · exp − Qc /3600 · Tc − − (2.12) Tf 2
where: Qc = circulating flow in front of the entry (pcu/h); nc = circle lane number; ne = entry lane number; Tc = critical gap; Tf = follow-up time; = minimum headway between the vehicles circulating in the circle. Therefore, according to Eq. (2.12), the capacity determination of roundabout entries is a function of the users’ behaviors, represented by the determination of the psycho-technical times Tc , Tf , and , as well as circulating traffic, the number of circle lanes, and the number of entry lanes. Setting the model under the German conditions on the basis of experimental data (Brilon), the psycho-technical times have been estimated as Tc = 4.1 s, Tf = 2.9 s, and = 2.1 s. Figure 2.8 shows the capacity behaviors determined using Eq. (2.12), as a function of the circulating traffic and of various geometric configurations of the roundabout (ne /nc ). Further developments of this capacity formula can be found in [9], where it is also recommended that Eq. (2.12) be used only in the case of roundabouts with a single-lane circle and single-lane entries. In the case of roundabouts that have circulatory roadways that can be used for vehicle traffic along two lines,4 the following relationship is used to determine the capacity C (pcu/h) of an entry:
4 The external diameter D ext must be between 40 and 60 m, and the central line of the circle must not be marked. In addition, the semi-practicable area around the central line must not be planned. The width of the circle must be a constant value of approximately 8, with a maximum of 10 m.
2.1
Capacity Calculation at Steady-State Conditions
33
Fig. 2.8 Capacity of roundabout entries according to the Brilon-Wu formula (HBS 2001)
C = 3600 ·
ne Q Tf · exp − c · Tc − Tf 3600 2
(2.13)
where: Qc = circulating flow in front of the entry (pcu/h); ne = parameter connected to the number of entry lanes; equal to 1 for singlelane entries and 1.4 for double-lane entries; Tc = critical gap = 4.3 s; Tf = follow-up time = 2.5 s. As an example, we determine the entry capacity for a geometric configuration (a four-legged roundabout, with a double-lane circle and double-lane entries) and with the traffic data already presented in Sects. 2.1.1 and 2.1.2. First, we determine the circulating flows in front of each entry (Sect. 2.1.1); then we use Eq. (2.13) to determine capacity C1 = 3600 · (ne1 /Tf ) · exp −Qc1 /3600 · (Tc − Tf /2) = = 3600 · (1.4/2.5) · exp −376/3600 · (4.3 − 2.5/2) = 1194 pcu /h and, similarly,
34
2 Capacity Evaluation
C2 = 3600 · (1.4/2.5) · exp −379/3600 · (4.3 − 2.5/2) = 1191 pcu/h C3 = 3600 · (1.4/2.5) · exp −284/3600 · (4.3 − 2.5/2) = 1291 pcu/h C4 = 3600 · (1.4/2.5) · exp −276/3600 · (4.3 − 2.5/2) = 1299 pcu/h
2.1.6 HCM 2000 Formula (USA) The HCM 2000 approach for evaluating the entry capacities C for roundabouts is limited to schemes with one lane in the circle and one lane at the entries and with circulating flow Qc not greater than 1200 pcu/h [1]. To evaluate C, the following equation is used: C= where:
Qc e−Qc Tc /3600 (pcu/h) 1 − e−Qc Tc /3600
(2.14)
Qc = circulating flow in front of the entry (pcu/h); Tc = critical gap (s); Tf = follow-up time (s). Since extensive experimental data on operating roundabouts in the US were not available when the latest edition of the HCM was published, the Manual procedure gives an interval of capacity values obtained with the following values of the parameters Tc and Tf . The upper bound of Eq. (2.14) is obtained with Tc = 4.1 s and Tf = 2.6 s, and the lower bound is obtained with Tc = 4.6 s and Tf = 3.1 s (See Fig. 2.9). With the traffic data of Table 2.4, the capacities evaluated with Eq. (2.14) and with the German formula (2.1) were compared for a roundabout with one lane in the circle and one at the entries. In this example the German formula is the following: C = 1218 − 0.74 · Qc (pcu/h)
(2.15)
In this case, we note that the average value between the lower and the upper bound of the capacity, evaluated by the HCM formula, is, in practice, the same as the value obtained by the German formula (2.15). Figure 2.10 shows a comparison of two capacity formulas, the HCM capacity formula (Eq. (2.14)) and the German formula (Eq. (2.12)). These two formulas are applied to a roundabout with single-lane entries and a single-lane circle. Equations (2.14) and (2.12) were evaluated using for Tc and Tf the values indicated by the German capacity formula (Tc = 4.1 s; Tf = 2.9 s). We can note (See Fig. 2.9) that the American formula overestimates capacity systematically when compared to the values obtained from the German formula. This result was observed when the same geometric and traffic conditions were used,
2.1
Capacity Calculation at Steady-State Conditions
35
Fig. 2.9 HCM 2000 capacity formula for different values of critical gap and follow-up time
Table 2.4 Traffic data and capacity value for a roundabout Capacity HCM 2000 formula
Entry
Maneuver
Volume
1
Right turn Straight Left turn Right turn Straight Left turn Right turn Straight Left turn Right turn Straight Left turn
22 208 164 59 432 44 187 266 38 40 134 13
2
3
4
Lower bound
Average
Qc
Upper bound
German formula (See Eq. 2.15)
185
1198
992
1095
1081
384
1023
834
929
933
640
834
667
751
744
348
1054
862
958
960
as well as when users had the same psycho-technical parameters, for values of Qc greater than 300–350 pcu/h.
36
2 Capacity Evaluation
Fig. 2.10 Comparison of the German HBS 2001 capacity formula and the HCM 2000 capacity formula
2.2 Exit and Circle Capacities Until now, no studies have been conducted that were specifically dedicated to the determination of roundabout exit and circle capacities. Regarding exits, field observations show that the capacity limit for each lane is in the range of 1200–1400 pcu/h.5 Regarding the circulatory roadway, we can use the values shown in Table 2.5 concerning observations about operating roundabouts in Germany just as an indication of the values that can be expected.
2.3 Consideration of Pedestrian Crosswalks In urban roundabouts, pedestrian crosswalks at legs reduce entry and exit capacities in proportion to the value of the pedestrian flow. In current technical practice, three calculation procedures are mainly used to determine the above-mentioned entry capacity reductions, i.e., the English 5
In Europe, for safety reasons, double-lane exits are rarely used.
2.3
Consideration of Pedestrian Crosswalks
37
Table 2.5 Circle capacity values according to empirical data from operating roundabouts in Germany Type of roundabout
Number of entry Circle capacity lanes [veh/h]
Roundabouts with single-lane circles (mini roundabouts and compact roundabouts) Compact roundabouts with double-lane circles Large roundabouts
1
1600
1 2 1 2
1600 1600 2000 2500
procedure (Marlow and Maycock), the German procedure (Brilon, Stuwe and Drews), and the French procedure (CETE de l’Ouest). All three of the procedures are valid only when the assumption is made that pedestrians at pedestrian crosswalks have priority over vehicular traffic. The English and French procedures are based on the same principles, since they both use the results of the mathematical queuing theory. The German procedure is based on the treatment of empirical data obtained from operating roundabouts. Regarding exits, specific formulas are not available at the present time, and, as will be discussed later, the same criteria that are used for exits are also used for entries.
2.3.1 Entry Capacities in Presence of Pedestrian Crosswalks 2.3.1.1 Marlow and Maycock Formula First, the capacity value Cap in the presence of only pedestrian flow Qped (Griffiths’ formula) is calculated [10]
Cap =
Qped Qped · β + (e
Qped ·α
where:
Qped = pedestrian flow (ped/s); β = C10 (s);
− 1) · (1 − e− Qped β)
· 3600
(2.16)
38
2 Capacity Evaluation
C0 = capacity with pedestrian and vehicular flows equal to zero (completely empty roundabout); α = B/vped = time necessary (s) to allow pedestrians to completely cross the pedestrian crosswalk, where B (m) is the width of the road at the pedestrian crosswalk, and vped (m/s) is the (mean) pedestrian flow speed. The width B, which characterizes each entry, must be defined separately for each entry according to the roundabout geometry. For vped , except for different direct determinations, one assumes vped = 0.5– 2.0 m/s, with the suggested default value of 1.4 m/s. Once known the value of Cap , entry capacity C/ped , which takes into account the pedestrian flow, is C/ped = C · M
(2.17)
where M is a reduction factor of the capacity C value (veh/h) of the entry considered in absence of the pedestrian flow provided by M=
Rn+2 − R Rn+2 − 1
(2.18)
Cap C
(2.19)
with R=
and “n” is equal to the number of vehicles that may be in the queuing area between the pedestrian crosswalk and the yielding line. The term “n,” which must be determined for each entry, is a function of the mean longitudinal size of the vehicles (equal to 5–6 m); to determine n, we consider all the entry lanes, e.g., in the case of a double-lane entry and a distance of 5 m between the pedestrian crosswalk and the yielding line, n = 2. Figures 2.11, 2.12, 2.13, and 2.14 show the relationship M = M(Qc ) for two urban roundabouts, i.e., a single-lane entry roundabout6 (Figs. 2.11 and 2.12) and a double-lane entry roundabout7 (Figs. 2.13 and 2.14). M was determined with the TRRL capacity procedure (Eq. (2.6)) and with the German HBS 2001 procedure (Brilon-Wu formula) (Eq. (2.12)). All the graphs were traced for increasing pedestrian flow values Qped from 100 to 800 ped/h with increments of 100 ped/h. The width of the pedestrian crosswalk B was assumed to be 3.5 m and 7.5 m, for singlelane entries and double-lane entries, respectively. Pedestrian speed was set to vped = 1.4 m/s. 6 The geometric parameter values used are as follows: D = 34 m; u = 7 m; e = 4 m; v = 3 m; = 7.5 m; = 35◦ ; r = 20.8 m. (The symbols for the parameter values are reported in Table 2.2.) 7 The geometric parameter values used are as follows: D = 56 m; u = 8 m; e = 8 m; v = 6.5 m; = 15.1 m; 35◦ ; r = 32.7 m. (The symbols for the parameter values are reported in Table 2.2).
2.3
Consideration of Pedestrian Crosswalks
39
Fig. 2.11 M = M(Qc ) relationship according to Marlow and Maycock for a single-lane roundabout with single-lane entries (capacity C determined by TRRL procedure)
Fig. 2.12 M = M(Qc ) relationship according to Marlow and Maycock for a single-lane roundabout with single-lane entries (capacity C determined with HBS 2001 procedure)
40
2 Capacity Evaluation
Fig. 2.13 M = M(Qc ) relationship according to Marlow and Maycock for a roundabout with double-lane entries (capacity C determined with TRRL procedure)
Fig. 2.14 M = M(Qc ) relationship according to Marlow and Maycock for a roundabout with double-lane entries (capacity C determined with HBS 2001 procedure)
2.3
Consideration of Pedestrian Crosswalks
41
2.3.1.2 Brilon, Stuwe and Drews Formula With this method, as with the method just illustrated in the previous section, the entry capacity C (determined with any procedure which doesn’t include pedestrian crosswalks) is reduced by means of a factor M that takes into account the effects of pedestrian crosswalks [11]: C/ped = C · M
(2.20)
M is given on the basis of entry configurations: – single-lane entry M=
1119.5 − 0.715 · Qc − 0.644 · Qped + 0.00073 · Qc · Qped 1069 − 0.65 · Qc
(2.21)
– double-lane entry
M=
1260.6 − 0.381 · Qped − 0.329 · Qc 1380 − 0.50 · Qc
(2.22)
where Qc = circulating flow in front of the entry (pcu/h); Qped = pedestrian flow crossing the leg (ped/h). Figures 2.15 and 2.16 show the relationship M = M(Qc ) for a single-lane entry roundabout (See Fig. 2.15) and a double-lane entry roundabout (See Fig. 2.16). The graphs were traced for increasing pedestrian flow values of Qped from 100 to 800 ped/h with increments of 100 ped/h. The equations that allow the determination of the reduction factor M may give unrealistic results, when used outside the existence intervals of the experimental measures. Thus, for example, in the case of single-lane entry roundabouts with a small volume of pedestrians (<100 ped/h), the formulas show that when there is a marginal increase in pedestrians Q ped , capacity also tends to increase. However, these circumstances do not invalidate the formula, but they demand careful application. 2.3.1.3 CETE de l’Ouest Formula Also with this procedure, entry capacity C is reduced by means of a factor F that takes into account the pedestrian flow [5] C/ped = C · F
(2.23)
42
2 Capacity Evaluation
Fig. 2.15 M = M(Qc ) relationship according to Brilon, Stuwe and Drews for a single-lane entry roundabout
Fig. 2.16 M = M(Qc ) relationship according to Brilon, Stuwe and Drews for a double-lane entry roundabout
2.3
Consideration of Pedestrian Crosswalks
43
with F = 1 − exp ( − k · Qd · β) · [1 − exp ( − Qped · T)]
(2.24)
where: Qd = disturbing traffic in front of the entry (pcu/s) (Qd must be determined according to the capacity formula chosen, e.g., Qd is given by Eq. (2.9) if one uses the GIRABASE procedure); Qped = pedestrian flow crossing the leg (ped/s); β =
1 (s); C0
C0 = capacity with pedestrian and vehicular flows equal to zero (completely empty roundabout); k = number of vehicles that may be in the area between the pedestrian crosswalk and the yielding line. The previous graphs were traced for increasing pedestrian flow values Qped from 100 to 800 ped/h with increments of 100 ped/h. Also Figure 2.17 shows the relationship F = F(Qc ) for increasing pedestrian flow values Qped from 100 to 800 ped/h with increments of 100 ped/h.
Fig. 2.17 F = F(Qc ) relationship according to the CETE formula
44
2 Capacity Evaluation
2.3.2 Exit Capacities in Presence of Pedestrian Crosswalks We can reasonably assume that a heavy pedestrian flow at an exit causes a capacity reduction of the exit. At first, these effects are generally and approximately determined with the formula by Marlow and Maycock (Sect. 2.3.1.1), even though recent studies have shown that, using the equations proposed by this method, the pedestrian influence on the roundabout exit capacity is overestimated. In conclusion, we wish to emphasize that the formulas relative to the determination of roundabout exit capacities demand careful application because they have not been specifically validated at this time.
2.4 Some Concluding Remarks on Capacity Formulations The differences between the formulas presented in the previous sections and between these formulas and other formulas from the literature are mainly caused by the following reasons, as are the discrepancies between their capacity estimates: – drivers’ behaviors at the intersection are due, among other things, to the extent of their experience with roundabouts on the road networks of their countries; – roundabout geometric standards vary from nation to nation, which makes them dissimilar from configurations that have the same number of circle lanes and entry lanes; – for the mixed traffic data used to determine and set calculation procedures. Some of these data are specific for certain environments, such as urban areas and rural areas, whereas other data are from contexts that are only generically similar to the environments from which the most significant samples come; – the environments where the roundabouts are built are different because of varying national urban and territorial features, even though different environments are called in the same way in the different languages; – there are correlations between the geometric variables and traffic values that can highlight or hide the role of some parameters during the experimental phase and statistical treatment of measures. Thus, these parameters may not be present in the capacity formulas. In some countries (as in Italy and Spain), a specific formula for the determination of circular intersection capacity has not yet been defined. Therefore, in these countries, it is essential to compare the different types of roundabouts present in the national standard (if one is available) and in foreign standards that address the same types of intersections. Thus, we can better choose the best calculation procedure for the case under examination, at least with respect to the conformity between the geometric standards and the environment. As we have just mentioned, different nations use different classifications and nomenclature, resulting in their standards not being fully compatible or interchangeable. (See, for example, Tables 2.6 and 2.7, which compare Italian, German, and Swiss definitions).
2.4
Some Concluding Remarks on Capacity Formulations
45
Table 2.6 Roundabout classification: Italian versus German nomenclatures Italian nomenclatures
German nomenclatures
Dext (m)
Dext (m)
Mini-roundabouts Compact roundabouts
14–25 25–40
13–24 26–60
Roundabouts “Rotary circulation” layout
40–50 >50
55–80 –
urban: 26–35 (single-lane circle) rural: 30–45 (double-lane circle) Urban and rural: 40–60 (double-lane circle)
Table 2.7 Roundabout classification: Italian versus Swiss nomenclatures Italian nomenclatures
Swiss nomenclatures
Dext (m)
Dext (m)
Mini-roundabouts
14–25
Small roundabouts
–
Compact roundabouts
25–40
Roundabouts Big roundabouts (Swiss Standard nomenclature) “Rotary circulation” layout (Italian Standard nomenclature)
40–50 –
14–20 (town centers, residential areas, urban areas) 19–25 (town centers, residential urban, and suburban areas) 25–35 (urban, suburban, and rural areas) – >35 (rural areas)
>50
–
46
2 Capacity Evaluation
On the other hand, one must make such comparisons in order to decide which capacity relationship to use. In fact, the users’ behaviors are not comparable when specific investigations are not available. Finally, for engineers of a country where a capacity formulation is not available, the selection of a capacity formula is not easy and must be seriously considered. However, when the engineers analyze the results obtained, they can use the comparison with the results derived from a different formula that has been deemed to be suitable for the case in question and/or compare the results with the ones of a traffic micro-simulation study of the roundabout.
2.5 Capacity Calculation at Saturation or Oversaturation Conditions of Entries If traffic demand at one or more entries equals or exceeds capacity, the system reaches a state characterized by flows coming from undersaturated entries equal to demand and from saturated or oversaturated entries equal to capacity. Now we recall that, an entry capacity depends on the circulating flows determined by entering flows coming from the other legs of the roundabout. Therefore, the entry capacity values calculated only on the basis of traffic demand at the intersection and without taking into account the saturation or oversaturation conditions of some entries are not correct. Thus, we will now present an iterative method for the determination of entering flows to a roundabout and the determination of the respective capacity values for saturated or oversaturated conditions at one or more entries. This procedure is illustrated for a four-legged roundabout, but it is always valid due to the mathematical nature of the problem. Traffic demand is known and is represented by the flow vectors [Qei ] and matrix PO/D , i.e., matrix MO/D (See Sect. 1.1). We proceed with successive calculation steps; for each step (k), we determine the flow values for each leg that can actually enter the roundabout. Thus, we obtain *(k) a vector [Qei ]; the iterative calculation converges very rapidly and ends when the *(k−1) *(k) same entering flows are reached for two successive steps, i.e., [Qei ] = [Qei ]. The values represent the volumes that can enter the roundabout examined, under the restraint that one or more legs must have demands equal to or exceeding their capacities (traffic rate ρi = Qei /Ci ≥ 1, i.e., reserve capacity RCi = Ci – Qei ≤ 0). At step (1) of the procedure, we assume that the disturbing traffic is zero in front (1) (1) of entry 1 (Qd1 = 0). Then, we determine the corresponding capacity C1 value *(1) and the flow Qe1 as the smaller of the calculated capacity value and traffic demand Qe1 . *(1) For entry 2, on the basis of flow Qe1 and traffic percentage matrix PO/D , (1) (1) we determine the disturbing traffic value (Qd2 ), the corresponding capacity C2
2.5
Capacity Calculation at Saturation or Oversaturation Conditions of Entries
47
*(1)
value, and the flow Qe2 , as the smaller of the calculated capacity value and traffic demand Qe2 . We proceed similarly for entries 3 and 4. Thus, for entry 3, we determine the (1) *(1) *(1) disturbing traffic (Qd3 ) (starting with Qe1 , Qe2 , and the traffic percentage matrix), (1) *(1) (1) capacity C3 and the flow Qe3 ; for entry 4, the disturbing traffic (Qd4 ) (starting *(1) *(1) *(1) (1) with Qe1 , Qe2 , Qe3 and the traffic percentage matrix), capacity C4 and the flow *(1) *(1) Qe4 . In the end, we obtain the vector [Qei ] of the entering flows at step (1). At step (2), we repeat the calculations starting with entry 1, and we determine (2) *(1) *(1) *(1) the disturbing traffic (Qd1 ) (starting with Qe2 , Qe3 , Qe4 and traffic percentage (2) *(2) matrix), capacity C1 , and the flow Qe1 . We proceed similarly, using an iterative method, for all of the other entries, until *(2) we obtain the vector [Qei ] of the entering flows at step (2). (k) Regarding, in particular, the determination of disturbing flow Qdi for the generic leg “i” at step (k) of the iterative method, it is worth emphasizing that it must be determined by means of the “most recent” values of the volumes that may enter (2) the roundabout from the other entry; thus, for example, for Qd2 , one uses the flow *(2) *(1) *(1) (2) *(2) *(2) *(1) values Qe1 , Qe3 , and Qe4 ; for Qd3 , one uses, Qe1 , Qe2 , and Qe4 , and so the process continues until it is completed. We proceed similarly, using an iterative method, for the following calculation *(3) *(k) steps. Thus, we obtain a succession of vectors of entering flows [Qei ], ..., [Qei ]; as already discussed, the calculation converges very rapidly and ends when the same *(k−1) ] = [Q*k entering volumes ([Qei ei ]) occur for two successive steps and the same *(k−1) *(k−1) *k disturbing flows ([Qdi ] = [Qdi ]) and capacities [Ci = C*k i ] are reached. Thus, we have determined the respective capacity values taking into account the entering flow values at the intersection and the saturation or oversaturation conditions of some of the entries. In particular, it is worth noting that, for the saturated or oversaturated legs, the capacity values Ci evidently coincide with those of the volumes Qei that may enter the roundabout.
2.5.1 A Worked Example We will now present an application of the procedure just illustrated. For the sake of simplicity, we use a capacity formula for which the disturbing traffic Qd consists only of the circulating flow Qc in front of the entries (Qd = Qc ). However, the procedure can be used without difficulty, even when Qd is also function of the exiting flow of the leg. (See, for example, Eq. (2.9)). Consider a four-legged roundabout for which the “traffic percentage matrix” is: ⎡
PO/D
0.00 ⎢ 0.24 =⎢ ⎣ 0.36 0.30
0.31 0.00 0.40 0.30
0.38 0.44 0.00 0.40
⎤ 0.31 0.32 ⎥ ⎥ 0.24 ⎦ 0.00
48
2 Capacity Evaluation
The traffic demand vector (pcu/h) is [Qei ] = 800 500 900 700 Therefore, the matrix O/D is ⎡
⎤ 0 248 304 248 ⎢ 120 0 220 160 ⎥ ⎥ [O/D] = ⎢ ⎣ 324 360 0 216 ⎦ 210 210 280 0 Among the roundabout capacity formulas available in the literature, we use the Brilon-Bondzio formula (2.1), as an example, C = A − B · Qc which is specialized for roundabouts with a single-lane circle and single-lane legs (A = 1218, B = 0.74). Starting with traffic data, we determine the circulating flow vector [Qci ] (pcu/h) (Eq. (1.12) in Chap. 1). [Qci ] = 850 832 528 804
The capacity vector (pcu/h), determined through the application of the abovementioned formula for each leg, is [Ci ] = 589 602 827 623
Comparing the traffic demand vector [Qei ] to the capacity vector [Ci ], we can notice that entries 1, 3, and 4 are at oversaturation conditions ([ρi ] = [Qi /Ci ] = [1.36 0.83 1.09 1.12]). As already discussed, the circulating flow values and, therefore, the capacity values just determined are not correct under such conditions. In fact, the actual flow entering the oversaturated legs is not equal to demand, but to capacity. Therefore, if one or more entries are saturated, we use the procedure calculation to determine a balanced situation. (1) (1) For entry 1, if we assume at step (1) that Qc1 = 0, then we obtain C1 = (1) ∗(1) (1) 1218 − 0.74 · Qc1 = 1218 − 0.74 · 0 = 1218 pcu/h and Qe1 = min (C1 ; Qe1 ) = min (1218; 800) = 800 pcu/h.
2.5
Capacity Calculation at Saturation or Oversaturation Conditions of Entries
49
This volume is divided into the following, on the basis of the traffic percentage matrix: ∗(1)
∗(1)
Q12 = P12 · Qe1 = 0.31 · 800 = 248 pcu/h ∗(1) ∗(1) Q13 = P13 · Qe1 = 0.38 · 800 = 304 pcu/h ∗(1) ∗(1) Q14 = P14 · Qe1 = 0.31 · 800 = 248 pcu/h For entry 2, we determine ∗(1)
(1)
∗(1)
Qc2 = Q13 + Q14 (1) C2
= 304 + 248 = 552 pcu/h (1)
= 1218 − 0.74 · Qc2 = 1218 − 0.74 · 552 = 810 pcu/h
∗(1)
(1)
Qe2 = min (C2 , Qe2 ) = min (810, 500) = 500 pcu/h This volume is divided into the following, on the basis of the traffic percentage matrix: ∗(1)
= P21 · Qe2
∗(1)
= P23 · Qe2
∗(1)
= P24 · Qe2
Q21 Q23 Q24
∗(1)
= 0.24 · 500 = 120 pcu/h
∗(1)
= 0.44 · 500 = 220 pcu/h
∗(1)
= 0.32 · 500 = 160 pcu/h
For entry 3, we determine ∗(1)
(1)
∗(1)
∗(1)
Qc3 = Q14 + Q21 + Q24 (1) C3 ∗(1) Qe3
= 1218 − 0.74 · =
(1) Qc3
(1) min (C3 , Qe3 )
= 248 + 120 + 160 = 528 pcu/h
= 1218 − 0.74 · 528 = 827 pcu/h
= min (827; 900) = 827 pcu/h
This volume is divided into: ∗(1)
= P31 · Qe3
∗(1)
= P32 · Qe3
∗(1)
= P34 · Qe3
Q31 Q32 Q34
∗(1)
= 0.36 · 827 = 298 pcu/h
∗(1)
= 0.40 · 827 = 331 pcu/h
∗(1)
= 0.24 · 827 = 198 pcu/h
For entry 4, we determine ∗(1)
(1)
∗(1)
∗(1)
Qc4 = Q21 + Q31 + Q32 (1) C4 (1) Qe4
= 1218 − 0.74 · =
(1) Qc4
(1) min (C4 ; Qe4 )
= 120 + 298 + 331 = 749 pcu/h
= 1218 − 0.74 · 749 = 664 pcu/h
= min (664; 700) = 664 pcu/h
This volume is divided into: ∗(1)
= P41 · Qe4
∗(1)
= P42 · Qe4
∗(1)
= P43 · Qe4
Q41 Q42 Q43
∗(1)
= 0.30 · 664 = 199 pcu/h
∗(1)
= 0.30 · 664 = 199 pcu/h
∗(1)
= 0.40 · 664 = 266 pcu/h
50
2 Capacity Evaluation
Table 2.8 Circulating flows [Qci ], capacity [Ci ], entering flows [Q*ei ], and matrix [M*O/D ] at step (1) of the iterative method Leg 1 2 3 4
(1)
(1)
*(1)
[Qci ]
[Ci ]
[Qei ]
0 552 528 749
1218 810 827 664
800 500 827 664
D
1
2
3
4
0 120 298 199
248 0 331 199
304 220 0 266
248 160 198 0
O 1 2 3 4
For step (1), Table 2.8 shows the circulating flow values [Qci ], capacity values [Ci ], entering flow values [Qei ∗ ], and matrix M∗O/D . The calculations proceed in an iterative way; in particular, it is worth noting that, as already discussed, the circulating flow Qci (k) in front of the generic leg “i” at step (k) of the procedure must be determined by means of the “most updated” values of the volumes that may enter the roundabout from the other legs, based on the previous calculation steps. For entry 1, we obtain ∗(1)
(2)
∗(1)
∗(1)
Qc1 = Q42 + Q43 + Q32 (2)
C1
= 199 + 266 + 331 = 796 pcu/h
(2)
= 1218 − 0.74 · Qc1 = 1218 − 0.74 · 796 = 629 pcu/h
∗(2)
(2)
Qe1 = min (C1 , Qe1 ) = min (629, 800) = 629 pcu/h and therefore ∗(2)
Q12
∗(2) Q13 ∗(2) Q14
∗(2)
= 0.31 · 629 = 195 pcu/h
∗(2) Qe1 ∗(2) Qe1
= 0.38 · 629 = 239 pcu/h
= P12 · Qe1 = P13 · = P14 ·
= 0.31 · 629 = 195 pcu/h
Similarly, for entry 2 we determine ∗(2)
∗(2)
∗(1)
(2) Qc2 = Q13 + Q14 + Q43 (2) C2 = ∗(2) Qe2 =
1218 − 0.74 · (2) min (C2 , Qe2 )
(2) Qc2
= 239 + 195 + 266 = 700 pcu/h
= 1218 − 0.74 · 700 = 700 pcu/h
= min (700; 500) = 500 pcu/h
and therefore ∗(2)
Q21
∗(2) Q23 ∗(2) Q24
∗(2)
= 0.24 · 500 = 120 pcu/h
∗(2) Qe2 ∗(2) Qe2
= 0.44 · 500 = 220 pcu/h
= P21 · Qe2 = P23 · = P24 ·
= 0.32 · 500 = 160 pcu/h
2.5
Capacity Calculation at Saturation or Oversaturation Conditions of Entries
51
For entry 3, we have ∗(2)
(2)
∗(2)
∗(2)
Qc3 = Q14 + Q21 + Q24 (2)
C3
= 195 + 120 + 160 = 475 pcu/h
(2)
= 1218 − 0.74 · Qc3 = 1218 − 0.74 · 475 = 867 pcu/h
∗(2)
(2)
Qe3 = min (C3 , Qe3 ) = min (867, 900) = 867 pcu/h and therefore ∗(2)
= P31 · Qe3 = 0.36 · 867 = 312 pcu/h
∗(2)
= P32 · Qe3 = 0.40 · 867 = 347 pcu/h
∗(2)
= P34 · Qe3
Q31 Q32 Q34
∗(2) ∗(2) ∗(2)
= 0.24 · 867 = 208 pcu/h
Finally, we determine for entry 4 ∗(2)
(2)
∗(2)
∗(2)
Qc4 = Q21 + Q31 + Q32 (2) C4 (2) Qe4
= 1218 − 0.74 · =
(2) Qc4
(2) min (C4 ; Qe4 )
= 120 + 312 + 346 = 778 pcu/h
= 1218 − 0.74 · 778 = 642 pcu/h
= min (642; 700) = 642 pcu/h
and therefore ∗(2)
Q41
∗(2) Q42 ∗(2) Q43
∗(2)
= 0.30 · 642 = 193 pcu/h
∗(2) Qe4 ∗(2) Qe4
= 0.30 · 642 = 193 pcu/h
= P41 · Qe4 = P42 · = P43 ·
= 0.40 · 642 = 257 pcu/h
For step (2), Table 2.9 shows the circulating flow values [Qci ], capacity values [Ci ], entering flow values [Qei ∗ ], and matrix M∗O/D . Proceeding similarly for the successive calculation phases, we obtain the values shown in Tables 2.10 and 2.11 (steps (3) and (4)).
Table 2.9 Circulating flows [Qci ], capacity [Ci ], entering flows [Q*ei ], and matrix M∗O/D at step (2) of the iterative method Leg 1 2 3 4
(2)
(2)
*(2)
[Qci ]
[Ci ]
[Qei ]
796 700 475 778
629 700 867 642
629 500 867 642
D
1
2
3
4
0 120 312 193
195 0 346 193
239 220 0 256
195 160 208 0
O 1 2 3 4
52
2 Capacity Evaluation
Table 2.10 Circulating flows [Qci ], capacity [Ci ], entering flows [Q*ei ], and matrix M*O/D at step (3) of the iterative method Leg 1 2 3 4
(3)
(3)
*(3)
[Qci ]
[Ci ]
[Qei ]
796 691 475 778
629 707 867 642
629 500 867 642
D
1
2
3
4
0 120 312 193
195 0 347 193
239 220 0 256
195 160 208 0
O 1 2 3 4
Table 2.11 Circulating flows [Qci ], capacity [Ci ], entering flows [Q*ei ], and matrix M*O/D at step (4) of the iterative method Leg 1 2 3 4
(4)
(4)
*(4)
[Qci ]
[Ci ]
[Qei ]
796 691 475 778
629 707 867 642
629 500 867 642
D
1
2
3
4
0 120 312 193
195 0 347 193
239 220 0 256
195 160 208 0
O 1 2 3 4
At calculation steps (3) and (4), the entering flow vectors [Q*ei ], circulating flow vectors [Q*ci ], and capacity vectors [Ci ] coincide.8 The iterative process converges, i.e., [Q∗ei ] = [629 500 867 642] is the flow vector that enters the roundabout taking into account the oversaturation of entries 1, 3, and 4; [C∗i ] = [629 707 867 642] is the capacity vector. As already pointed out, we can notice that, for oversaturated entries, the flow values that may enter the roundabout are equal to the capacity values of the respective entries. On the other hand, for entry 2, which is at undersaturation conditions, the entering flow value is equal to traffic demand.
2.6 Calculation Procedures for Simple Capacity and Total Capacity As already pointed out in Sect. 1.2, roundabout performance indicators as a whole are, in technical practice, “simple capacity” SC and “whole capacity” or “total capacity” TC. The determinations of SC and TC are not straightforward, and they require the development of suitable computational procedures.
8
The iterative results and the final results are all given approximated integers.
2.6
Calculation Procedures for Simple Capacity and Total Capacity
53
For the sake of simplicity, but without narrowing our general aims, we will now present these procedures for a four-legged roundabout with an assigned traffic demand (initial condition) represented by the flow vector [Qei ] and by matrix PO/D (i.e., matrix MO/D ). To calculate simple capacity means to determine a specific condition that is characterized by equivalence between demand and capacity values for at least one of the entries. This is done by means of a uniform increase in entering flow from an initial traffic state. 1) we calculate the disturbing flows (Qdi ) for each entry starting with the demand vector [Qei ] and traffic percentage matrix PO/D ; 2) for each entry, we express the entering flow and disturbing flow values under saturation conditions as the multiplication of traffic demand by a suitable multiplier (different for each entry), i.e., Q*ei = δi Qei and Q*di = δi Qdi. Thus, the generic capacity formula used for the calculation can be written as δi Qei = fi (δi Qdi ). Solving these equations, we obtain the values of the multipliers δi (one for each entry) of traffic demands at legs when these same legs are saturated; 3) we multiply demand vector [Qei ] by the smallest multiplier that we have found (corresponding to the first one that reaches saturation), and, thus, we determine simple capacity SC. To determine total capacity, we use the following iterative method: (1)
(1)
(1)
1) we assume an arbitrary, first-attempt, entering flow vector [Qei ] = [Qe1 Qe2 (1) (1) Qe3 Qe4 ]; (1) 2) for entry 1 we determine the disturbing traffic Qd1 as a function of the first (1) attempt flow vector [Qei ] and matrix [Pij ], and, by the capacity formula used, (2) we determine a new value of entering flow 1 Qe1 ; 3) for entries 2, 3, and 4, we proceed similarly. First, we determine the disturbing flows (using the most “updated” entering flows for each leg, i.e., those obtained with the application of the iterative method). Then, by means of the capacity formula, we determine the new entering flow values, obtaining for them a second (2) vector [Qei ]; 4) we repeat the calculation process until we obtain, for two successive steps, two (k−1) (k) entering flow vectors that are equal, i.e., when [Qei ] = [Qei ]. The entering flow vector thus determined represents the number of the entering vehicles from each leg when all the entries are simultaneously saturated. The roundabout total capacity value TC is the sum of all these flows. Finally, it is worth noting that total capacity is a function only of the traffic percentage matrix PO/D , and, therefore, it can be determined starting with an arbitrary demand vector. In other words, with equal percentage distribution, total capacity
54
2 Capacity Evaluation
vector is univocal, independently of the demand vector used for the first calculation step (first-attempt demand vector). Contrary to total capacity, it should be noted that simple capacity is function of both traffic demand and its percentage distribution (vector [Qei ] and matrix PO/D ).
2.6.1 A Worked Example We will now present a example calculation of simple capacity and total capacity. Consider a four-legged roundabout with a traffic percentage matrix PO/D of: ⎡
PO/D
0.00 ⎢ 0.19 =⎢ ⎣ 0.63 0.19
0.15 0.00 0.15 0.74
0.75 0.24 0.00 0.07
⎤ 0.10 0.57 ⎥ ⎥ 0.22 ⎦ 0.00
We use the Brilon-Bondzio (2.1) formula as the capacity formula. C = A − B · Qc This formula is specialized for roundabouts with a single-lane circle and singlelane legs (A = 1218, B = 0.74). Traffic demand vector (pcu/h) is equal to [Qei ] = [160 100 240 200]
2.6.1.1 Simple Capacity Calculation To calculate simple capacity, we first determine the circulating flow vector [Qci ] (pcu/h). (Eq. (1.12) in Chap. 1). [Qci ] = 198 150 92 206 For each entry, we look for the initial flow multiplier that causes saturation at the entry by means of the relationship δi Qei = A–B · δi Qci , which, when applied to the case under examination, becomes δ1 · δ2 · δ3 · δ4 ·
160 100 240 200
= = = =
1218 − 0.74 · 1218 − 0.74 · 1218 − 0.74 · 1218 − 0.74 ·
δ1 · δ2 · δ3 · δ4 ·
198 150 92 206
2.6
Calculation Procedures for Simple Capacity and Total Capacity
55
from which we have δ1 δ2 δ3 δ4
= = = =
3.97 5.77 3.95 3.46
Multiplying traffic demand [Qei ] by the smaller multiplier (δ4 = 3.46), we determine the entering flow values that correspond to the first roundabout congestion (SC) event for a uniform increase in the flows (i.e., simple capacity) [Qei ] = [SC] (pcu/h): (SC)
[Qei ] = [SC] = [553 346 829 691] Under traffic conditions for which entering flow values are equal to simple capacity, we can also determine the circulating flow values (pcu/h) in front of the entries. (SC)
[Qci ] = [684 518 318 712] capacities (pcu/h) [C(SC) ] = [712 834 983 691] and reserve capacities (pcu/h) [RC(SC) ] = [159 489 153 0] It is worth noting that, under this traffic condition, the reserve capacity is zero for the leg (in the case under examination, leg n◦ 4) that reaches the saturation state first. 2.6.1.2 Total Capacity Calculation To calculate total capacity, we can randomly choose the first-attempt, entering flow vector. For the case under examination, we assume (1)
[Qei ] = [100 220 300 300] Then, we determine the circulating volume in front of entry 1 (1)
(1)
(1)
Qc1 = (P42 + P43 ) · Qe4 + P32 · Qe3 = (0.74 + 0.07) · 300 + 0.15 · 300 = 288 and, by the capacity formula, a new entering flow value at leg 1 (in pcu/h) (2)
(1)
Qe1 = 1218 − 0.74 · Qc1 = 1218 − 0.74 · 288 = 1005 pcu/h
56
2 Capacity Evaluation
We proceed similarly for entries 2, 3, and 4. We use the “most updated” entering traffic flows coming from the various legs to calculate the conflict flows: (1)
(2)
(1)
(1)
(2)
(2)
(1)
(2)
(2)
Qc2 = (P13 + P14 ) · Qe1 + P43 · Qe4 = (0.75 + 0.10) · 1005 + 0.07 · 300 = 875 pcu/h (2) (1) Qe2 = 1218 − 0.74 · Qc2 = 1218 − 0.74 · 875 = 570 pcu/h Qc3 = (P24 + P21 ) · Qe2 + P14 · Qe1 = (0.57 + 0.19) · 570 + 0.10 · 1005 = 534 pcu/h (2) (1) Qe3 = 1218 − 0.74 · Qc3 = 1218 − 0.74 · 534 = 823 pcu/h Qc4 = (P31 + P32 ) · Qe3 + P21 · Qe2 = (0.63 + 0.15) · 823 + 0.19 · 570 = 750 pcu/h (2) (1) Qe4 = 1218 − 0.74 · Qc4 = 1218 − 0.74 · 750 = 663 pcu/h Thus, we obtain a second entering flow vector (pcu/h) (2)
[Qei ] = [1005 570 823 663] We proceed similarly, and we obtain (3)
[Qei ] = [729 725 756 680] (4)
[Qei ] = [727 726 756 680] (5)
[Qei ] = [727 726 756 680] The iterative process converges very rapidly. The entering flow vectors determined for steps (4) and (5) are equal, and they represent the number of vehicles that each leg is able to serve when all the legs are saturated. The sum of all these flows provides the roundabout total capacity TC, which is 2888 pcu/h.
References 1. Highway Capacity Manual HCM 2000, Special Report n. 209, T.R.B., Washington DC, 2000. 2. Brilon, W., Wu, N., Bondzio, L., Unsignalized Intersections in Germany – A State of the Art 1997, Proceedings of the Third International Symposium on Intersections Without Traffic Signals, pp. 61–70, Portland, Oregon, USA, July 1997. 3. Brilon, W. (Editor), Intersections Without Traffic Signals, Vol. II, Springer-Verlag, Berlin, 1991. 4. Kimber, R. M., The Traffic Capacity of Roundabouts, TRRL Laboratory, Report 942, Crowthorne, Berkshire, 1980.
References
57
5. Louah, G., Panorama critique des modeles français de capacité des carrefours giratoires, Actes du séminaire international “Giratoires 92,” Nantes, France, 14–16 Octobre 1992. 6. Guichet, B., Roundabouts in France: Development, Safety, Design and Capacity, Proceedings of the Third International Symposium on Intersections Without Traffic Signals, pp. 100–105, Portland, Oregon, USA, July 1997. 7. Guichet, B., Roundabouts in France and New Use, TRB Transportation Research, National Roundabout Conference Proceedings 2005, Vail, Colorado, USA, 22–25 May 2005. 8. CERTU, Carrefours urbains Guide, Lyon, France, 1999. 9. Merkblatt für die Anlage von Kreisverkehren, FGSV Verlag GmbH, Köln, Germany, 2006. 10. Marlow, M., Maycock, G., The Effect of Zebra Crossings on Junction Entry Capacities, Report SR 742, Transport and Road Research Laboratory (TRRL), Crowthorne, Berkshire, England, 1982. 11. Brilon, W., Stuwe, B., Drews, O., Sicherheit und Leistungsfähigkeit von Kreisverkehrsplätzen, Institute for Traffic Engineering, Ruhr Universität, Bochum, Deutschland, 1993.
Chapter 3
Waiting Phenomena at Steady State and Non-steady State Conditions
To calculate waiting times, queue lengths, and the number of vehicles in the system,1 it is possible to consider roundabout entry lanes as channels of an ordinary system examined by means of the mathematical queuing theory (Fig. 3.1). Therefore, for the elements shown in Fig. 3.1 and for the specific case being considered here, we have that2 : – the arrival and departure processes, which are made up of a events that are generally represented by random instants (flows of events), are characterized on the basis of their respective traffic count laws [1]; – the service point is located on the entry yielding lines. The service method (service mechanism) consists of waiting for a time gap in the traffic stream flowing on the circulatory roadway that is sufficient to insert one or more vehicles in sequence into the roundabout (gap-acceptance [2]). On the basis of this service mechanism, the service time proprieties Ts are established in probabilistic terms (probabilistic distribution). – the waiting organization (queue discipline) requires that the users be served on the basis of their arrival time in the system (sometimes called “first come, first served” (FCFS), but more frequently referred to as “first in, first out” (FIFO); – the number of queuing users can be high (unlimited longitudinal capacity of the channel). Once in the queue, a vehicle cannot leave it without entering and exiting from the roundabout. In the remaining part of this chapter, first, we will review some results of stochastic and deterministic theory about waiting phenomena. Then, we will present a general heuristic criterion that allows the evaluation of the system state variables at any operational conditions of the entries (i.e., undersaturation, saturation, and oversaturation). 1 We 2
recall that the definitions of these state variables were given in Sect. 1.3. Examining the elements shown in Fig. 3.1, the queuing theory terms are reported in brackets.
R. Mauro, Calculation of Roundabouts, DOI 10.1007/978-3-642-04551-6_3, C Springer-Verlag Berlin Heidelberg 2010
59
60
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Fig. 3.1 Schematic diagram of a queuing system
The above-mentioned method uses transition relationships between results obtained with the probabilistic and deterministic approaches. Finally, in the next chapter, the criteria and the methods are applied to the study of roundabouts that have time-dependent traffic demands; in particular, we will consider roundabouts with traffic peaks that occur between two steady-state periods of flow.
3.1 Some Results of Probabilistic Analysis of Queues From the probabilistic theory of queuing, we will now review some definitions and fundamental relationships for the applications. We define traffic intensity or degree of saturation as the ratio ρ of traffic demand Qe at an entry to capacity C of the same entry3 ρ=
Qe C
(3.1)
If E[Ts ] is the mean of service time Ts , given the meaning of C and Ts , we can obtain E[Ts ] =
1 C
(3.2)
Recalling Eq. (1.17) of Chap. 1, by which we defined reserve capacity (RC = C − Qe) , from Eqs. (3.1) and (3.2), we have ρ=1−
RC C
(3.3)
Therefore, the entry condition can be characterized in terms of RC or of ρ as shown in Table 3.1. 3
For simplicity notation, in this chapter we omit the subscripts to the flows and capacities relative to the generic entry “i”.
3.1
Some Results of Probabilistic Analysis of Queues
61
Table 3.1 Reserve capacity RC, degree of saturation ρ and entry states conditions Undersaturated entry
Saturated entry
Oversaturated entry
RC > 0 ρ<1
RC = 0 ρ=1
RC < 0 ρ>1
If we indicate the average time spent in the system with E[ws ], using Eq. (1.20) of Chap. 1 (ws = wc + Ts ) and Eq. (3.2), above, we have the following equation for the propriety of the mean as a linear operator: E[ws ] = E[wc ] + E[Ts ] = E[wc ] +
1 C
(3.4)
In addition, to calculate the average number E[Ls ] of vehicles waiting in the system, it is necessary to know the relationship that exists between E[Ls ], Qe , and E[ws ] under steady-state conditions (first formula by Little): E[Ls ] = Qe · E[ws ]
(3.5)
Equation (3.5) can briefly be accounted for as follows. During the waiting time ws of a generic vehicle, the number of vehicles that enter the system is equal to: Ls = Qe · ws
(3.6)
Under conditions of statistical equilibrium, Qe results are time-independent, so we can average the terms of Eq. (3.6) to obtain Eq. (3.5). A relationship that is homologous to Eq. (3.6) can be written for the queue length Lc , obtaining Lc = Qe ·wc .
3.1.1 Some Remarkable Results We will now present some results of the queuing theory relative to unsignalized intersections at statistical equilibrium (steady-state conditions of the system). First, we must characterize the demand flow Qe probabilistically and specify the service mechanism on which the determination of service time Ts depends. In this connection, it is possible to use for roundabouts – as for any type of unsignalized intersections – different methods for evaluating Ts . For their detailed descriptions, see [2], where each method for calculating Ts is presented, along with the formulas for the averages E[ws ] and E[Ls ]. In current practice, however, the most widespread service mechanism is the one described below. Consider A and B to be the first two vehicles in the system. Assume that vehicle A uses an interval t equal to or slightly greater than the critical gap Tc for its maneuver. Vehicle B’s time Ts occurs between the moment that it arrives at the
62
3
Waiting Phenomena at Steady State and Non-steady State Conditions
yielding line and the moment it enters the circulatory roadway. Thus, for B, a time elapses that is equal to the sum of: the interval t–Tc + the possible successive gaps between the circulating vehicles rejected by B + the critical gap Tc . This sum is the service time Ts of vehicle B. If Ts is determined in this way, it is not possible to model, for more than one vehicle, a service that is provided with continuity inside the same time interval and in sequence, i.e. without interruptions of the entering process into the circulatory roadway. With the service procedures just recalled, the well-known P-K relationships (from the authors Pollaczeck-Khinchine) [3] relative to queuing theory can be easily used. According to these relationships: 2
E[ws ] = E[Ts ] +
Qe · (E[Ts ] + VAR[Ts ]) 2 · (1 − Qe · E[Ts ])
(3.7)
where E[Ts ] and VAR[Ts ] are the mean and the variance of service time, respectively. If we assume a Ts distribution according to the Erlang function with a parameter k, these two variables can be obtained from the following equations: ekQc Tc −
i=0
E[Ts ] = Tc + Qc
k i=0
(k + 1) · ekQc Tc −
k+1 i=0
VAR[Ts ] = kQ2c
k−1 i=0
k
(kQc Tc )i i!
(3.8)
(kQc Tc )i i!
(kQc Tc )i i!
+ (E[Ts ] − Tc )2
(3.9)
(kQc Tc )i i!
The parameter k of the Erlang function must be calculated on the basis of the traffic flow rate Qc , which can be approximated according to the indications shown in Table 3.2. Remember that Qc for a roundabout is the circulating flow in front of the entry considered (See Fig. 1.2 in Chap. 1).
Table 3.2 Parameter k value of the Erlang probability law versus of circulating flow Qc Flow rate Qc [veh/h]
K value
0–500 501–1000 1001–1500
1 2 3
3.1
Some Results of Probabilistic Analysis of Queues
63
The relationship shown in Table 3.2 was determined experimentally. If Qc (expressed as hourly volume) is known, the relationship can be used to determine k as an integer value for intervals of Qc . Equations (3.7), (3.8), and (3.9) are valid for Poissonian arrivals into the secondary stream Qe and service time Ts distributed in any way (depending on Tc and on the characteristics of the main stream Qc , represented by the value selected for k). Evaluating E[ws ] by means of Eq. (3.7), the average number E[Ls ] of vehicles in the system is given by Eq. (3.5) on the basis of which: 2
E[Ls ] = Qe · E[ws ] = Qe · E[Ts ] +
Q2e · (E[Ts ] + VAR[Ts ]) 2 · (1 − Qe · E[Ts ])
(3.10)
The non-approximated determination of the probability distribution Ls is not easy to represent in closed form. However, it is worth noting that doubling E[Ls ] generally provides a high enough percentile (Ls,90 ; Ls,95 ) estimation of Ls for the applications. Finally, we recall that the results and the considerations outlined so far to calculate E[ws ] are always relative to the simple case of only two conflicting flows (Qe and Qc ) or to schemes that can be assimilated to it (See Fig. 1.2 in Chap. 1). A simple, concise deduction of Eqs. (3.7) and (3.10) is reported in [3]. Equations (3.7) and (3.10) are now specified for two cases that are important for practical applications. First of all, we simplify the notations in Eq. (3.7) as: Ts = s; E[Ts ] = s; VAR [Ts ] = V[s] Therefore, we rewrite Eq. (3.7) as: E[ws ] = s +
Qe · (s2 + V[s]) 2 · (1 − Qe · s)
(3.11)
which, with Eqs. (3.1) and (3.2), becomes: V[s] ρ· s+ s E[ws ] = s + 2 · (1 − ρ)
(3.12)
From Eq. (3.10), we have for the average number E[Ls ] of vehicles in the system:
E[Ls ] = Qe · s +
Q2e · (s2 + V[s]) 2 · (1 − Qe · s)
(3.13)
64
3
Waiting Phenomena at Steady State and Non-steady State Conditions
which, again for Eqs. (3.1) and (3.2), is: ρ2
· 1+
V[s]
s2 2 · (1 − ρ)
E[Ls ] = ρ +
(3.14)
As we will see for some cases, with Eqs. (3.12) and (3.14) we have a straightforward characterization of the expressions of E[ws ] and E[Ls ]. 3.1.1.1 Poissonian Arrivals and Exponential Service Times Under this assumption, since the probability density function fTs (t) of the service time Ts , is exponential, we have the following relationship between mean and variance: V[s] = s2
(3.15)
Then, Eq. (3.12) becomes:
E[ws ] = s +
ρ s+
s2 s
=
2(1 − ρ)
s 2s − 2ρs + 2ρs = 2(1 − ρ) (1 − ρ)
(3.16)
Finally, using Eq. (3.2), we have: E[ws ] =
1 C · (1 − ρ)
(3.17)
From Eq. (3.14), we have for E[Ls ]: ! ρ2 E[Ls ] = ρ +
· 1+
s2 s2
2(1 − ρ)
" =
ρ − ρ2 + ρ2 ρ = 1−ρ 1−ρ
(3.18)
3.1.1.2 Poissonian Arrivals and Deterministic Service Times This occurs when the service is regular, i.e., each waiting vehicle spends the same time at the head of the queue to enter the circulatory roadway. This means that V[s] = 0
(3.19)
Then, taking into account Eq. (3.2), Eq. (3.12) becomes: E[ws ] = s +
ρ·s 1 2−ρ = · 2 · (1 − ρ) C 2 · (1 − ρ)
(3.20)
3.1
Some Results of Probabilistic Analysis of Queues
65
Table 3.3 Average values of state variables according to statistical equilibrium Queue E[wc ]
System E[Lc ]
E[ws ]
E[Ls ]
Poissonian arrivals; exponential service times
ρ C · (1 − ρ)
ρ2 (1 − ρ)
1 C · (1 − ρ)
ρ (1 − ρ)
Poissonian arrivals; constant service times
ρ 2 · C · (1 − ρ)
ρ2 2 · (1 − ρ)
2−ρ 2 · C · (1 − ρ)
2ρ − ρ2 2 · (1 − ρ)
From Eq. (3.14), we can write for E[Ls ]: E[Ls ] = ρ +
ρ2 2ρ − 2ρ2 + ρ2 2ρ − ρ2 = = 2(1 − ρ) 2(1 − ρ) 2(1 − ρ)
(3.21)
Starting once again with Eqs. (3.7) and (3.10), expressions for the averages E[wc ] and E[Lc ] can be easily derived for the waiting time in the queue and queue length. The expressions of the average values so far recalled are shown in Table 3.3.
3.1.2 Concluding Remarks To conclude this section, Fig. 3.2 shows the graphs of Eqs. (3.17) and (3.18). We can note that the behavior of the average time spent in the system E[ws ] and of the average number of users in the system E[Ls ] are similar (as is predictable on the basis of Eq. (3.5)). They are both characterized by a knee, beyond which, the
Fig. 3.2 Averages E[ws ] and E[Ls ] versus degree of saturation ρ (Poissonian arrivals; exponential service times; one-queue system)
66
3
Waiting Phenomena at Steady State and Non-steady State Conditions
gradients grow rapidly when traffic intensity ρ increases, and a vertical asymptote appears when ρ = 1 lim E[ws ] = ∞
(3.22)
lim E[Ls ] = ∞
(3.23)
ρ→1
ρ→1
We have similar behaviors for the queue, for E[wc ], and for E[Lc ]. We can easily note that, if E[ws ] and E[Ls ] are expressed as a function of reserve capacity, we can use Eqs. (3.17) and (3.18) to obtain the curves shown in Fig. 3.3, where the asymptote coincides with the ordinate axis. As we will soon see with an example, this behavior of E[ws ] and E[Ls ] derives directly from the two basic conditions of stationariness, i.e., from the assumption that the entering traffic demand remains constant indefinitely. In addition, this asymptotic behavior of E[ws ] and E[Ls ] is not only characteristic of queuing systems with Poissonian arrivals and exponential service times. In fact, all the queuing systems that may be modeled by Eqs. (3.7) and (3.10) show a similar behavior for E[ws ] and E[Ls ], as can be easily seen from Eqs. (3.12) and (3.14) for ρ which tends to be unity. Consider an entry that is systematically saturated, i.e., Qe (t) = C(t) and Qe (t+t) = Qe (t) for any t. For example, Qe is equal to 1440 veh/h = 0.40 veh/s, i.e., on average, 0.4 arriving vehicles are recorded for each second. C = Qe indicates that there is, on average, 0.4 vehicle served (in the case vehicles entering the circulatory roadway of a roundabout) for each second used for the service. For T = 10000 s, about 4000 arrivals are recorded. Due to the random variations of the system, the arrivals of 4000 vehicles will not be distributed uniformly over T, which means that sometimes there will be a queue, and sometimes there will not be a queue.
Fig. 3.3 Averages E[ws ] and E[Ls ] versus Reserve Capacity RC (Poissonian arrivals; exponential service times; one-queue system)
3.2
Deterministic Analysis of Queues
67
For the same reasons, the time gaps used for the service will have different values and will not be localized uniformly in T = 10000 s of system observation. In addition, their overall duration will be shorter than T. For example, if we assume that the overall duration is 9000 s, the number of arrivals into the roundabout is therefore of 9000·C = 9000·0.4 = 3600 vehicles. In conclusion, at the end of the T during which 4000 vehicles approach the intersection along the leg considered, and 3600 vehicles enter the roundabout, there would be 400 vehicles waiting in the queue for entry, which is an estimation of E[Ls ]. It is now straightforward to understand that, when all the other conditions are equal, if T increases indefinitely, even E[Ls ] tends to increase indefinitely (Eq. (3.23)). This is consistent with the presence of the asymptote that occurs when ρ = 1, as predicted by Eq. (3.17) (See Fig. 3.2). From Eq. (3.5), the presence of the asymptote when ρ = 1 is also justified for E[ws ].
3.2 Deterministic Analysis of Queues The analyses are based on the cumulative time value t of arrivals (at legs) A(t) and departures D(t). A departure from the queue occurs when a vehicle waiting in second position moves to the head of the queue, close to the yielding line. When there is no queue, the term departure in this context also refers to the travel of a vehicle from the head of the traffic stream to the waiting position to enter the intersection. In conclusion (See Fig. 3.4), a departure is recorded when a vehicle crosses the imaginary line (a–a) that marks the upstream limit to the service position. Since traffic counts are discrete, A(t) and D(t) are step functions. Instead of these, the respective continuous approximations are generally used (fluid approximations, Fig. 3.5). The continuous approximation derives from the continuous description (or continuous at intervals) of traffic demand Qe (t) and entry capacity C(t). From the definition of volume, the arrival rate is therefore equal to dA(t) = Qe (t) ∀t dt
(3.24)
With the deterministic approach, Qe (t) and C(t) are not subject to random variations. Under this assumption: – the determinations of Qe and C coincide with the mean values Qe (t) = E[Qe (t)];
C(t) = E[C(t)] ∀t
(3.25)
68
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Fig. 3.4 Priority system and service counter
– for the cumulative rate of departures it is straightforward to put: a)
dD(t) = QD (t) = Qe (t) dt
t ∈ (ti ; tj )
Fig. 3.5 Fluid approximation of the cumulative arrival and departure counts
(3.26)
3.2
Deterministic Analysis of Queues
69
when, during an entire generic interval (ti ; tj ), the entry is undersaturated (e.g. Qe (t) < C(t)) and at the initial instant there is no queue at the entry (e.g. Lc (ti ) = 0) b)
dD(t) = QD (t) = C(t) dt
t ∈ (ti ; tj )
(3.27)
if during the entire generic interval (ti ; tj ), the entry is saturated or oversaturated (e.g. Qe (t) ≥ C(t)) or if at the initial instant the queue L0 is present at entry (e.g. Lc (ti ) = L0 ). If Lc (ti ) = L0 and C(t) > Qe (t), tj = f(L0 ; C(t)). Once the behaviors of Qe (t) and C(t) are known, A(t) and D(t) are obtained by integration from Eqs. (3.24), (3.26), and (3.27). Starting with the curves A(t) and D(t) (See Fig. 3.6), the main indicators of the waiting phenomenon can be calculated directly. The length of the queue Lc (t) at a generic instant t is obtained as Lc (t) = A(t) − D(t)
(3.28)
Equation (3.28) expresses the preservation law of vehicles at legs, on the basis of which the arriving vehicles can only leave the queue or stay in it (A(t) = D(t) + Lc (t)). As the queue method for entering the intersection is of the FIFO type (See Sect. 1.3), the horizontal distance d between the curves A(t) and D(t) is the waiting time in the queue wc for the vehicle that arrived at time t: d = wc (t)
Fig. 3.6 Cumulative arrival count A(t) and cumulative departure count D(t)
(3.29)
70
3
Waiting Phenomena at Steady State and Non-steady State Conditions
It follows that the area S(aba) included between A(t) and D(t) is the total waiting time in the queue Wc , i.e., regarding all the users who have entered the queue. The calculation of Wc allows the evaluation of the mean waiting time wc as a ratio between Wc and a suitable number of users represented by vehicles relative to the pre-fixed time Tq = ti –tf : #tf w ¯c =
Wc #tf ti
Qe (t)dt
=
[A(t) − D(t)]dt
ti
#tf
(3.30) Qe (t)dt
ti
The use of the functions A(t) and D(t) also allows, as we will later see with a worked example, the determination of the time Td inside which the effects of the peak traffic can be observed, when this same peak is ended. In the above-mentioned example, as it generally occurs in the analysis of the performance indices of intersections, the demand Qe and capacity C are modeled with step functions. With Eqs. (3.24), (3.26), and (3.27), broken lines are obtained by which the calculation (and implementation of the results which it gives) of the cumulative arrival and departure values are very straightforward due to the use of elementary geometrical considerations. So far, all that has been said about departures from the queue can be repeated for departures from the system (See Fig. 1.5, Chap. 1). In fact, the waiting time in the queue wc , as we have already seen, is equal to the horizontal distance between the cumulative counts A(t) and D(t) (See Fig. 3.6); in addition, the relationship between the time spent in the system ws , service time Ts , and queuing time wc is expressed by Eq. (1.20) from Chap. 1.
ws = wc + Ts = wc +
1 C
(3.31)
The cumulative count Ds (t) of the departures from the system is therefore obtained instant by instant by moving the value of D(t) horizontally by a value equal to Ts = 1/C.
3.2.1 A Worked Example Consider an entry with a variable traffic demand. Capacity during the observation period has a constant behavior. To simplify the notation, the subscript for traffic demand, i.e., “e,” is omitted.
3.2
Deterministic Analysis of Queues
71
Initial Data The arrival flow rate at the legs is Q1 = 1600 veh/h from 6:45 to 7:00 AM. In addition, it results that Q1 < C = 2000 veh/h. During the period from 7:00 to 7:15 AM, the arrival rate increases to Q2 = 2400 veh/h, so Q2 >C. From 7:15 to 7:30 AM, demand becomes Q3 = 2200 veh/h, and Q3 > C. Finally, between 7:30 and 7:45 AM the arrival rate decreases to Q4 = 1200 veh/h, so Q4 < C. The volumes Q1 , Q2 , Q3 , and Q4 and capacity C (rate of flowing vehicles) shown in Fig. 3.7a cause (relationships (3.24), (3.26), and (3.27)), the curves of the cumulative arrival values and of the cumulative departure values of the queue as functions of time. The curves are represented, respectively, by the line indicated as “arrivals” and by the “departures” line shown in Fig. 3.7b on the basis of the relationships: Cumulative value of demand (arrivals) = Q · t
(3.32)
Cumulative count of departures from the queue = C · t
(3.33)
Number of Arriving Vehicles The number of arriving vehicles between 6:45 and 7:00 AM is: A1 = Q1 · t1 = 1600 veh/h · (0.25 h) = 400 veh This number is equal to the area B1 of the bar diagram shown in Fig. 3.7a. Similarly, for each of the remaining analysis periods, the cumulative counts A2 , A3 , and A4 are determined for a total of 1850 vehicles in the 45 min considered. Time of the Beginning of the Congestion Congestion begins exactly when demand exceeds capacity (i.e., when the entry is oversaturated), that is, when Q2 > C. This event is recorded at 7:00 AM. Congestion Duration (Tq = tf – ti ) During the periods t2 and t3 , the arriving flows exceed capacity C, while the queue disappears in the interval t4 (See Fig. 3.7b) and remains for the duration Td (part of t4 ): Tq = t2 + t3 + Td
(3.34)
72
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Fig. 3.7 Demand flow rate, cumulative arrival count and cumulative departure count at leg
In addition, in the unknown time Tq , the following vehicles must be cleared: (Q2 –C)·t2 ; (Q3 –C)·t3 ; and Q4 ·Td ; that is in all: (Q2 − C) · t2 + (Q3 − C) · t3 + Q4 · Td As in Td , the queue Lmax , present at the end of t3 , is cleared
3.2
Deterministic Analysis of Queues
73
(Q2 − C) · t2 + (Q3 − C) · t3 + (C − Q4 ) · Td = 0 and, therefore, Td =
(Q2 − C) · t2 + (Q3 − C) · t3 C − Q4
which, substituted into Eq. (3.34), gives (Q2 −C)·t2 + (Q3 −C)·t3 C−Q4
Tq = t2 + t3 +
= 0.25 + 0.25 +
=
(2400−2000)·0.25 + (2200−2000)·0.25 2000−1200
= 0.69 h = 41.4 min
In other words, the queue that began at 7:00 o’clock AM disappears at 7:41:24 AM. Maximum Number of Queuing Vehicles Lmax The maximum number of queuing vehicles is recorded at the end of period t3 and is equal to the cumulative value of demand rate that was not cleared during the period t2 + t3 . Lmax = (Q2 − C) · t2 + (Q3 − C) · t3 = = (2400 − 2000) · 0.25 + (2200 − 2000) · 0.25 = 150 veh Maximum Duration of the Individual Delay wcmax This delay refers to the vehicle that arrives at the end of the period t3 , that is at 7:15 AM, and is equal to (See Fig. 3.7b) wcmax =
Q2 ·t2 +Q3 ·t3 C
− t2 − t3 =
3 −C)·t3 = (Q2 −C)·t2 +(Q = C = 0.075 h = 4.5 min
Q2 ·t2 +Q3 ·t3 −C·t2 −C·t3 C
=
(2400−2000)·0.25+(2200−2000)·0.25 2000
=
Total Duration of the Delay Wc Wc is represented by the area S of the cumulative diagram (Qe ; C). 2 )·t2 2 )·t3 3 −C)·t3 ] +$ (Q2 t2 −Ct + %[(Q2 −C)·t2 +(Q · t3 + Wc = S = (Q2 t2 −Ct 2 2 2 + =
[(Q2 −C)·t2 +(Q3 −C)·t3 ] (Q2 t2 −Ct2 )·(t2 +t3 ) 2
2
+
(Q2 −C)·t2 +(Q3 −C)·t3 C−Q4
(Q2 −C)·t2 +(Q3 −C)·t3 2
= $[2400·0.25−2000·0.25]·(0.25+0.25) + 2 · 0.25 +
=$ t3 +
(Q2 −C)·t2 +(Q3 −C)·t3 C−Q4
%
(2400−2000)·0.25+(2200−2000)·0.25 · 2 %
(2400−2000)·0.25+(2200−2000)·0.25 2000−1200
= 57.81 (h veh)
=
74
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Number of Delayed Vehicles NR NR = C · Tq = 2000 · 0.69 = 1380 veh Mean Delay for Each Vehicle Affected by the Disturbance wc wc =
Wc 57.81 = = 0.041 h ∼ = 2.5 min NR 1380
Average Length of the Queue Lc
Lc =
Wc 57.81 ∼ = = 84 veh Tq 0.69
3.2.2 Some Remarkable Results The deterministic approach is used to analyze two important cases for the applications of practical interest. 3.2.2.1 First Remarkable Case and a Worked Example Assume (See Fig. 3.8) that at the initial instant t0 of system observation, Lc (t0 ) = Lc0 vehicles are present in the queue. Traffic demand Qe (t) and capacity C(t) starting with t0 are systematically assumed to be constant (Qe (t) = Qe and C(t) = C), with Qe > C (oversaturated entry). At the instant t, the cumulative arrival and departure counts are, on the basis of Eqs. (3.24) and (3.27), A(t) = A(t0 ) + Qe · T
(3.35)
D(t) = D(t0 ) + C · T
(3.36)
Since A(t0 ) – D(t0 ) = Lc0 , we can use Eqs. (3.28), (3.35), and (3.36) to produce Lc (t) = Lc0 + (Qe − C) · T
(3.37)
3.2
Deterministic Analysis of Queues
75
Fig. 3.8 Traffic demand, capacity, and cumulative arrival and departure values regarding the first remarkable case
Recalling Eq. (3.1), Eq. (3.37) becomes Lc (t) = Lc0 + (ρ − 1) · C · T
(3.38)
By definition, the number of vehicles in the system Ls (t) (See Sect. 1.3) is Ls (t) = Lc (t) + 1
(3.39)
76
3
Waiting Phenomena at Steady State and Non-steady State Conditions
With Eq. (3.38), Eq. (3.39) becomes Ls (t) = Ls0 + (ρ − 1) · C · T
(3.40)
Ls (t0 ) = Ls0 = Lc0 +1 be the number of users in the system at the initial instant t0 . The calculation of the average waiting time in the queue wc and of the time spent in the system ws for the vehicles that arrived between t0 and t = t0 + T is straightforward, i.e., the last vehicle in a queue with a length of Lc0 waits in the queue for a time wc (t0 ) that, on the basis of Eq. (3.29), is equal to the segment bf in Fig. 3.8: wc (t0 ) = bf =
Lco C
(3.41)
Again, with Eq. (3.29), the waiting time in the queue wc (t) of the last vehicle that reach the queue at t = t0 + T is (See Fig. 3.8) wc (t) = cd =
Lc (t) C
(3.42)
and therefore, for Eq. (3.38), wc (t) =
Lc0 + (ρ − 1) · T C
(3.43)
The mean wc (See Sect. 1.3, Eq. (1.22)) is wc (t0 ) + wc (t) 2
(3.44)
1 Lc0 Lc0 Lc0 (ρ − 1) · T + + (ρ − 1) · T = + 2 C C C 2
(3.45)
wc = then, with Eqs. (3.41) and (3.43), wc =
The average value wc is equal to the value of the waiting time of the vehicle that arrives in the middle of the interval T. Average value wc can also be obtained on the basis of Eq. (3.30) as a ratio between the area S(bcdfb) and the number of vehicles that arrived in the waiting queue between t0 and t = t0 + T: wc =
S(bcdfb) Qe · T
(3.46)
Since the relationship between the average waiting time in the queue wc and the time spent in the system ws is expressed by Eq. (3.4), since the number of users in the system is Ls0 = Lc0 +1 (Eq. (3.39) with t = t0 ), from Eq. (3.45), we have:
3.2
Deterministic Analysis of Queues
ws =
77
Lc0 (ρ − 1) · T 1 Ls0 (ρ − 1) · T + + = + C 2 C C 2
(3.47)
To give an example of the application of the relationship obtained, assume that Lc0 = 4 veh at t0 . Starting with t0 , we also obtain Qe = 1100 veh/h = 0.306 veh/s, C = 980 veh/h = 0.272 veh/s, and, therefore, ρ = Qe /C = 1.125. Now, we want to calculate the maximum length of the queue after T = 10 min = 600 s, the average waiting time in the queue wc , and the time spent in the system ws for the vehicles that arrived during the peak period T. With Eq. (3.38) we obtain Lc (t) = Lc0 + (ρ − 1) · C · T = 4 + (1.125 − 1) · 0.272 · 600 = 24.4 veh From Eq. (3.45) wc =
Lc0 (ρ − 1) · T 4 (1.125 − 1) · 600 + = + = 52.21 s C 2 0.272 2
From Eq. (3.47) ws =
Ls0 (ρ − 1) · T 5 (1.125 − 1) · 600 + = + = 55.88 s C 2 0.272 2
For Eq. (3.4), the difference ws − wc must be equal to the service time Ts = 1/C = 3.67 s; in fact, we have ws − wc = 55.88 − 52.21 = 3.67 s. The just obtained value wc can be also calculated with the evaluation of the area S(bcdfb) (See Fig. 3.8) by applying Eq. (3.46). We have $
Lc0 C
% $ · T−
Lc0 C
% + Tcd · C =
1 2
·
=
1 2
·
=
1 2
· [14.71 + 89.71] · [(600 − 14.71) + 89.71] · 0.272 ∼ = 9585 veh · s
$
+
Lc (t) C
S(bcdfb) =
4 0.272
+
24.4 0.272
% $ · 600 −
4 0.272
% + 89.71 · 0.272 =
and, therefore, with Eq. (3.46) wc =
S(bcdefb) 9585 = = 52.21 s Qe · T 0.306 · 600
Sometimes [4], the average waiting times wc and ws are expressed in a slightly different way from Eqs. (3.45) and (3.47), because Lc0 and Ls0 are meant to be the number of vehicles that a user who arrives at the initial instant t0 finds already in the queue or in the system, respectively. Under this assumption, the initial length of the queue Lc (t0 ) is then equal to Lc (t0 ) = Lc0 + 1; similarly, for Ls (t0 ), the result is Ls (t0 ) = Ls0 + 1. Repeating the deductive procedure that led to Eqs. (3.45) and (3.47), we can easily obtain
78
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Lc0 + 1 (ρ − 1) · T + C 2 Ls0 + 1 (ρ − 1) · T ws = + C 2
wc =
(3.48) (3.49)
3.2.2.2 Second Remarkable Case and Worked Examples With the criteria presented so far, it is also easy to calculate the variables of the waiting phenomenon associated with a traffic demand and capacity behavior of the type shown in Fig. 3.9, i.e., at the end of the peak period T (i.e., the instant tp = t0 + T), the entry oversaturation ends and the result is, starting with tp , systematically Qe1 < C1 and C1 > C. For Eqs. (3.24) and (3.27), the cumulative arrival count A(t) is the line (bcd); the cumulative departure count D(t) is the line (afd). The maximum queue length Lcmax and the maximum number of vehicles in the system Lsmax are obtained at the end of the traffic peak (at tp = t0 + T) by means of Eqs. (3.38) and (3.40), respectively. Figure 3.9 shows how the waiting phenomenon continues, diminishing little by little after the peak period T, for another interval Td . Only at the end of Td (at the instant td ), the queue is completely cleared, and we can write Lc (td ) = A(td ) – D(td ) = 0. Td is obtained from the relationship (See Fig. 3.9) Td =
Lcmax C1 − Qe1
(3.50)
From Eqs. (3.38) and (3.1), Eq. (3.50) can be written Td =
Lc0 + (ρ − 1) · C · T (1 − ρ1 ) · C1
(3.51)
To calculate the waiting time wc (t0 ) = wc0 = bg, Eq. (3.41) is valid, while wc (tp ) = wcmax can be easily obtained from Fig. 3.9. wc max = ce =
Lc max C1
(3.52)
and, therefore, with Eq. (3.38) wc max =
Lc0 + (ρ − 1) · C · T C1
(3.53)
To calculate the average waiting time wc for the vehicles that arrived during T, Eq. (3.30) is used.
3.2
Deterministic Analysis of Queues
79
Fig. 3.9 Traffic demand, capacity, and cumulative arrival and departure values for the second remarkable case
In this case, the total waiting time Wc is equal to the area S(bcfgb): c0 )·T Wc = Lcmax2·wcmax + (Lcmax +L − 2 1 = 2 · (A + B − D)
1 2
· Lc0 ·
Lc0 C
=
(3.54)
80
3
Waiting Phenomena at Steady State and Non-steady State Conditions
where, expressing the terms of the second member of Eq. (3.54) with Eqs. (3.38) and (3.53), A, B, and D are A=
[Lc0 + (ρ − 1) · C · T]2 C1
B = 2 · Lc0 · T + (ρ − 1) · C · T2 D=
L2c0 C
(3.55) (3.56) (3.57)
Having put E = Qe · T
(3.58)
with Eq. (3.30) we obtain wc =
1 · (A + B − D) 2·E
(3.59)
To calculate the average waiting time in the queue wcq relative to all the delayed vehicles, i.e., vehicles that arrived in the queue during the interval Tq = T + Td , we must take into account the total waiting time in the queueW∗c , which is equal to the area S(bcdfgb). Thus we have (See Fig. 3.9) W∗c =
Lcmax · Td Lc0 (Lcmax + Lc0 ) · T 1 + − · Lc0 · 2 2 2 C
(3.60)
Equation (3.30), with Eqs. (3.38) and (3.51), is written 1 · (A∗ + B − D) 2
(3.61)
[Lc0 + (ρ − 1) · C · T]2 (1 − ρ1 ) · C1
(3.62)
W∗c = where A∗ =
while B and D are given by Eqs. (3.56) and (3.57), respectively. If we put now E∗ = Qe · T + Qe1 · Td
(3.63)
that, for Eq. (3.51), becomes E∗ = Qe · T + Qe1 · on the basis of Eq. (3.30), wcq is
Lc0 + (ρ − 1) · C · T (1 − ρ1 ) · C1
(3.64)
3.2
Deterministic Analysis of Queues
81
Fig. 3.10 Discharge process of peak traffic effects with the queue final length Lc1 greater than zero
wcq =
1 · (A∗ + B − D) 2 · E∗
(3.65)
If the effects of the peak period are considered to end when Lc1 vehicles are present in the queue (See Fig. 3.10), Eq. (3.50) of Td is modified as T d =
Lcmax − Lc1 Lcmax − Lc1 = C1 − Qe1 C1 · (1 − ρ1 )
(3.66)
because, by Eq. (3.1), ρ1 = Qe1 /C1 . The relationship that gives the total waiting time Wc in this case is written taking into account the area S(bcdeghb) (See Fig. 3.10), and it is 1 · (A∗∗ + B − D) 2
(3.67)
[Lc0 + (ρ − 1) · C · T]2 − L2c1 (1 − ρ1 ) · C1
(3.68)
W∗∗ c = where A∗∗ =
82
3
Waiting Phenomena at Steady State and Non-steady State Conditions
since Lcmax is always given by Eq. (3.38) and T d by Eq. (3.66); again, B and D are given by Eqs. (3.56) and (3.57), respectively. Putting
E∗∗ = Qe · T + Qe1 · Td = Qe · T + Qe1 ·
[Lc0 + (ρ − 1) · C · T] − Lc1 (1 − ρ1 ) · C1
(3.69)
with Eq. (3.69), for Eq. (3.30), the average waiting time in the queue w cq is w cq =
1 · (A∗∗ + B − D) 2 · E∗∗
(3.70)
where B and D are again given by Eqs. (3.56) and (3.57), respectively. Finally, the calculation of the average time spent in the system ws for vehicles that arrived during the peak T is conducted, on the basis of Eq. (3.31), adding the mean value Ts of the service time Ts to the mean wc . Ts is evaluated taking into account that (See Fig. 3.9) during the fraction T of T, Ts = 1/C, while in T , Ts = 1/C1 . Therefore, for Ts as the weighted mean, we have Ts =
1 C
· C · T +
1 C1
· C1 · T
C · T + C1 · T
=
T + T C · T + C1 · T
(3.71)
where T = T − (Lc0 /C)
(3.72)
T = wcmax = Lcmax /C1
(3.73)
In conclusion, for Eq. (3.31) we have, with Eqs. (3.59) and (3.71) ws = Ts + wc =
T + T 1 + (A + B − D) C · T + C1 · T 2·E
(3.74)
With the same procedure that led to Eq. (3.71), we can obtain the mean values Tsq and T sq of service time relative to all the delayed vehicles, in the case of complete clearance of the queue at the end of the interval Td (Eq. (3.50)) and in the case of a queue Lc1 greater than zero at the end of T d (Eq. (3.66)), respectively. Thus, we have the following relationship:
Tsq =
1 C
· C · T +
1 C1
· C1 · T d
C · T + C1 · Td
=
T + Td C · T + C1 · Td
(3.75)
3.2
Deterministic Analysis of Queues
83
where T and Td are given by Eqs. (3.72) and (3.50), respectively. T sq =
1 C
· C · T +
1 C1
· C1 · T d
C · T + C1 · T d
=
T + T d C · T + C1 · T d
(3.76)
where T and T d are given by Eqs. (3.72) and (3.66), respectively. In conclusion, on the basis of Eq. (3.31), with Eqs. (3.65) and (3.75), we have the average time spent in the system wsq = Tsq + wcq =
T + T d 1 + (A∗ + B − D) C · T + C 1 · T d 2 · E∗
(3.77)
Similarly, again on the basis of Eq. (3.31), with Eqs. (3.70) and (3.76), we obtain:
w sq = T sq + w cq =
T + T d 1 + (A∗∗ + B − D) C · T + C1 · T d 2 · E∗∗
(3.78)
As an example, we consider the case of an entry for which there are Lc0 = 4 veh queuing at time t0 . We refer here to Fig. 3.9. In T = tp – t0 = 15 min = 900 s, for the arrival demand and capacity we have, Qe = 1100 veh/h = 0.306 veh/s and C = 980 veh/h = 0.272 veh/s, respectively, and, therefore, ρ = Qe /C = 1.125. Starting with tp , we have Qe1 = 850 veh/h = 0.236 veh/s and C1 = 1010 veh/h = 0.281 veh/s, and, therefore, ρ1 = Qe 1 / C1 = 0.840. At the end of the peak period T, the maximum queue length Lcmax (Eq. (3.38)) is recorded Lc max = Lc0 + (ρ − 1) · C · T = 4 + (1.125 − 1) · 0.272 · 900 = 34.6 veh and the maximum waiting time in the queue wcmax (Eq. (3.52)) is wc max =
Lc max 34.6 = = 123.13s ∼ = 2 min C1 0.281
The average waiting time in the queue wc for the vehicles that arrived during the peak period T is evaluated by calculating, with Eqs. (3.55), (3.56), and (3.57), the quantities A, B, and D, which must be inserted into Eq. (3.59): A=
34.62 [Lc max ]2 = = 4260.36 veh · s C1 0.281
B = 2·Lc0 ·T+(ρ−1)·C·T2 = 2·4·900+(1.125−1)·0.272·9002 = 34740 veh ·s
84
3
Waiting Phenomena at Steady State and Non-steady State Conditions
D=
wc =
L2c0 42 = = 58.82 veh · s C 0.272
1 1 · (A + B − D) = · (4260.36 + 34740 − 58.82) = 2 · Qe · T 2 · 0.306 · 900
= 70.70s = 1.18min The average time spent in the system relative to the vehicles that arrived during the traffic peak, ws , is obtained with Eq. (3.74), once that by Eqs. (3.72) and (3.73) the periods T and T are evaluated and, therefore, with them, according to Eq. (3.71), the mean service time Ts T = T − (Lc0 /C) = 900 − (4/0.272) = 885.3 s T = wc max = 123.13 s
Ts =
T + T 885.29 + 123.13 = = 3.66 s C · T + C1 · T 0.272 · 885.29 + 0.281 · 123.13 ws = Ts + wc = 3.66 + 70.70 = 74.36 s
The duration of the interval Td , where the effects are visible once the peak ends, by Eq. (3.50), is Td =
Lc max 34.6 = = 768.89 s C1 − Qe1 0.281 − 0.236
Having determined the value of Td , it is possible to calculate E∗ (Eq. (3.63)) and, once A∗ has been evaluated by Eq. (3.62), Eq. (3.65) can be used to obtain the average waiting time in the queue wcq relative to all the vehicles that arrived during the period (Tq = T + Td ), (See Fig. 3.9). Therefore, we have E∗ = Qe · T + Qe1 · Td = 0.306 · 900 + 0.236 · 768.89 = 456.86 veh
A∗ =
[Lc max ]2 34.62 = = 26627.22 veh · s (1 − ρ1 ) · C1 (1 − 842) · 0.281
and by them, with B and D calculated above, 1 ∗ wcq = 2·E ∗ · (A + B − D) = = 67.10 s = 1.12 min
1 2·456.86
· (26627.22 + 34740 − 58.82) =
3.2
Deterministic Analysis of Queues
85
The average time spent in the system wsq relative to the vehicles that arrived during Tq is given by Eq. (3.77), once known (with T = 885.29 s and Td = 768.89 s), the mean service time Tsq (Eq. (3.75)) Tsq =
T + Td 885.29 + 768.89 = = 3.62 s C · T + C1 · Td 0.272 · 885.29 + 0.280 · 768.89
Then we have wsq = Tsq + wcq = 3.62 + 67.10 = 70.72 s = 1.18 min If we consider that the effects of the peak period ends in presence of a queue Lc1 at the end of the period T d (which is still affected by the demand peak in T, Fig. 3.10), the extension of T d is obtained by Eq. (3.66). If we assume Lc1 = 3 veh and use the data from this example, we obtain T d =
Lc max − Lc1 34.6 − 3 = = 702.22 s C1 − Qe1 0.281 − 0.236
The calculation of the average waiting time in the queues w cq and w sq requires the evaluation of A∗∗ (Eq. (3.68)), E∗∗ (Eq. (3.69)), and T sq (Eq. (3.76)). With the above-mentioned relationships, we have A∗∗ =
L2c max − L2c1 (1 − ρ1 ) · C1
=
34.62 − 32 = 26427.05 veh · s (1 − 0.840) · 0.281
E∗∗ = Qe · T + Qe1 · T d = 0.306 · 900 + 0.236 · 702.22 = 441.12 veh
T sq =
T + T d 885.29 + 702.22 = = 3.62 s C · T + C1 · T d 0.272 · 885.29 + 0.281 · 702.22
with B and D already calculated, by Eq. (3.70), we obtain w cq =
1 2·E∗∗
· (A∗∗ + B − D) =
1 2·441.12
· (26427.05 + 34740 − 58.82) =
= 69.26 s = 1.15 min Knowing w cq and T sq , from Eq. (3.78) we obtain w sq = T sq + w cq = 3.62 + 69.26 = 72.88 s = 1.21 min
86
3
Waiting Phenomena at Steady State and Non-steady State Conditions
3.2.3 Further Expressions of the Average Waiting Times The expressions of the mean times wc , wcq , and w cq given in the previous sections were obtained from the waiting times only for the vehicles that arrived during the period T and in the following period that was still affected by the peak demand. Now, if we want to take into account all the vehicles included in the queue starting with the beginning t0 of the peak period, it is obvious that the term D in Eqs. (59), (65), and (70) must not be subtracted from the terms A and B in brackets. In addition, the values of E, E∗ , and E∗∗ must be added to the value Lc0 of the length of the possible queue that may be present at the entry at the instant t0 . Thus, we have 1 · (A + B) 2 · (E + Lc0 ) 1 wcq = · (A∗ + B) ∗ 2 · (E + Lc0 ) 1 w cq = · (A∗∗ + B) 2 · (E∗∗ + Lc0 ) wc =
(3.79) (3.80) (3.81)
Even for the expressions of times spent in the system, Eqs. (3.74), (3.77), and (3.78) are modified to take into account the possible presence of the queue at the instant t0 . Also, the means of the service times must, in fact, be added to Eqs. (3.79), (3.80), and (3.81). In the expressions (3.71), (3.75) and (3.76) of the above-mentioned means, the interval T must be substituted for total T (See Fig. 3.10) to account for the time (T–T ) used to serve the vehicles that form Lc0 : T + T 1 + (A + B) C · T + C1 · T 2 · (E + Lco ) T + Td 1 = + (A∗ + B) C · T + C1 · Td 2 · (E∗ + Lc0 )
ws = Ts + wc = wsq = Tsq + wcq
w sq = T sq + w cq =
T + T d 1 + (A∗∗ + B) ∗∗ C · T + C1 · T d 2 · (E + Lc0 )
(3.82) (3.83) (3.84)
Frequently [5], as indicators of the effects of traffic demand peaks, average conventional values of waiting times are used (in the queue or in the system) obtained as a ratio between total delays 1 ∗ (A + B) 2 1 = (A∗∗ + B) 2
W∗c = W∗∗ c
and only the vehicles that arrived during the peak period T, equal to Qe ·T.
(3.85) (3.86)
3.2
Deterministic Analysis of Queues
87
Using the same criterion, the mean service time must be calculated only in relation to the above-mentioned vehicles. Thus, we have Tsq = T sq =
T + T C · T + C1 · T
(3.87)
In conclusion, by Eqs. (3.85), (3.86), and (3.87), we have 1 · (A∗ + B) 2 · Qe · T 1 = · (A∗∗ + B) 2 · Qe · T
wcq = w cq wsq = Tsq + wcq = w sq = T sq + w cq =
(3.88) (3.89)
T + T 1 + · (A∗ + B) C · T + C1 · T 2 · Qe · T
(3.90)
T + T 1 + · (A∗∗ + B) C · T + C1 · T 2 · Qe · T
(3.91)
3.2.4 Concluding Remarks The results illustrated in the previous sections were based on the following assumptions: – the cumulative arrival A(t) and departures D(t) counts are generally approximated with a continuous process (fluid approximation) that, in the case of a succession of periods with constant traffic demand and capacity, are continuous at intervals (piecewise-linear); – during a period of constant traffic demand Qe , the vehicles reach the leg at constant intervent times τA = 1/Qe ; – during a period when capacity C is constant, the vehicles depart from the queue (or the system) at constant time rates equal to a τD = 1/C. The fluid approximation becomes even more valid when the system is congested. We recall that congestion is recorded if the entry is oversaturated, saturated, or undersaturated but starting with an initial state with a long queue. During the congestion, the queue length is greater than unity, and the waiting times are greater than the mean service time. Under these conditions, the values of the discontinuities of A(t) and D(t) (equal to unity when there is an arrival or a departure – Fig. 3.5) are small with respect to the mean values in the time of the same functions, i.e., it follows that we have, for A(t) and D(t), small increases relative to the progression of the processes. Similarly, we have small differences between the continuous approximations and the starting relationships (step functions).
88
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Fig. 3.11 Generic behavior of the average time spent in the system versus the value of saturation degree
The regularity assumptions of the arrival and departure processes become even more plausible when the traffic conditions become heavier, since, at these conditions, the random changes of the above-mentioned processes tend to diminish, with the consequence that the prevailing effects are given by the means, while the fluctuations around them tend to diminish. 4 Finally, it is worth noting that the solutions provided by the deterministic approach belong to the type of time-dependent solutions of the queuing theory. In fact, unlike the probabilistic solutions, they take into account both the finite extension of the congestion periods of the system and the prolongation of peak effects. As an example of a time-dependent solution, Fig. 3.11 shows the generic behavior of the average time spent in the system (provided by Eq. (3.49)) as the peak interval T value increases when Ls0 = 0. We will present further time-dependent solutions (not deterministic) in the following section.
3.3 Time-Dependent Solutions and Waiting Phenomena As we have already pointed out in the previous sections, the solutions provided by the probabilistic approach are valid at a steady-state condition. 4
In queuing theory, the fluid approximation (deterministic approximation) has a wider meaning than the one just recalled, since it involves the consideration of the relationship between the arrival and departure process realizations and the respective levels (means affected by the previous values) and the mean time values. About this aspect, see [3] and [7].
3.3
Time-Dependent Solutions and Waiting Phenomena
89
For a real system, these conditions are only partially possible. The above-mentioned solutions are considered to be acceptable approximations for the entries “i” if the time Ti during which traffic demands Qei and capacity Ci may be assumed to be constant is sufficiently wide and if the entry is undersaturated in Ti and the traffic intensity ρi = Qei /Ci is suitably smaller than unity. 5 In conclusion, by the probabilistic results it is not possible to study situations in which Qei is time-dependent, and it can also exceed capacity Ci in a specific peak period. Regarding the deterministic approach, in the observations presented in previous Sect. 3.2.4, we pointed out that the results become more reliable as the entry becomes more oversaturated (in presence or absence of queues at the initial instant of system observation), i.e., with the increase of ρi > 1. In fact, under this circumstance, the random effects on the state variables (queues and waiting times) become smaller, and the expected values of the same stochastic state variables near deterministic means. In other words, for the conditions at which the entry is oversaturated with ρi > 1 but not with ρi >> 1, the deterministic approach cannot take into account the random effects of the evolution of the waiting phenomenon in the real system, leading to the systematic underestimation of the values of the state variables realizations. Then, to describe the non-steady situations, i.e., variability Qei and/or oversaturation of the entry, but with ρi not sufficiently greater than unity, the two approaches – probabilistic and deterministic – so far described are not suitable. In mathematical queuing theory, thorough discussions are available for these cases. Some of them are capable of solving particular problems in a precise way, others in approximate ways, but none of them is simple enough for practical applications [3], [6], [7]. That is why, in the analysis of a system under steady-state conditions, we generally use heuristic criteria of transition from statistical solutions to deterministic solutions. Thus, we obtain (for the calculation of queue lengths, number of users in the system, and waiting times) relationships in which traffic intensity ρ and the time T relative to the duration of system observation or the peak period appear as independent variables. More precisely, a state variable δ (e.g., Lc, Ls , wc , or ws ) is calculated as average <δ> for a fixed value of the interval T, suitably combining its expected value E[δ] under statistical equilibrium conditions (steady-state conditions) with its mean deterministic value δ, so that (See Fig. 3.12) the oblique asymptote of the relationship <δ> = <δ(ρ)> coincides with the deterministic function that yields δ. The determination of <δ> = <δ(ρ)> may be performed in different ways. The two following criteria are usually used.
5
For the value of Ti , see the indication included in footnote 4 of Chap. 1 and the following Chap. 4.
90
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Fig. 3.12 Averages of E[δ], δ , and <δ> versus traffic intensity ρ
Given generic equal values of E[δ], δ, and <δ> (E[δ] = δ = <δ>), let x be the distance between E[δ] and the vertical asymptote for ρ = 1, and let y be the distance between the curve <δ> = <δ(ρ)> (to be determined) and the half-line δ = δ(ρ) that is its oblique asymptote. Then, we can write x=y
(3.92)
x:1 = y:ρd
(3.93)
or, alternatively,
In terms of traffic intensity, for Eq. (3.92), we have (See Fig. 3.12) 1 − ρe = ρd − ρT
(3.94)
ρe = ρT − (ρd − 1)
(3.95)
thus
3.3
Time-Dependent Solutions and Waiting Phenomena
91
If we assume, instead, Eq. (3.93) we have 1 − ρe =
ρd − ρT ρd
(3.96)
ρT ρd
(3.97)
thus ρe =
Starting with Eq. (3.95) or Eq. (3.97) and choosing explicit ρe and ρd values as functions of E[δ] and δ, we have, for an assigned T, two (different) expressions of <δ> = <δ(ρ)>. Given the heuristic nature of the deductive criterion of the behavior of <δ> under time-dependent conditions (i.e., non-steady), there are no specific reasons to prefer one of the Eqs. (3.92) and (3.93) over the other (or other equations that may be used to obtain the transition <δ> between E[δ] and δ with an asymptotic behavior to δ). In the following sections, in accordance with the main trend in the technical literature [4, 5], Eq. (3.92) will be used to obtain time-dependent solutions for the average number of users in the system Ls , the queue length Lc , and the waiting times ws and wc .
3.3.1 Time-Dependent Solutions for the Number of Users in the System and in the Queue. A Worked Example The behaviors of traffic demand Qe and capacity C are described in the previous Sect. 3.2.2.1 (first remarkable case) (See Fig. 3.8). For the average E[Ls ] of the number of users in the system under a steady condition, Eq. (3.18) is used E[Ls ] =
ρ 1−ρ
(3.18)
and, for the average Ls of the same state variable6 at the end of the period T of system observation, Eq. (3.40) is used Ls (t) = Ls0 + (ρ − 1) · C · T
(3.40)
Now, to obtain the transition curve < Ls > between E[Ls ] and Ls , we use Eq. (3.92) or Eq. (3.95) ρe = ρT − (ρd − 1) 6
(3.95)
We recall that, in the deterministic approach, the value of a state variable coincides with the average value calculated in the same instant and with the percentiles, regardless of their order.
92
3
Waiting Phenomena at Steady State and Non-steady State Conditions
We recall (See Fig. 3.12) that ρe , ρd , and ρT are the values of traffic intensities that result in E[Ls ] = Ls =< Ls >
(3.98)
Therefore, with ρ = ρd and Eq. (3.40), we can use Eq. (3.98) to obtain ρd =
Ls − Ls0 < Ls > −Ls0 +1= +1 C·T C·T
(3.99)
and with ρ = ρe and Eq. (3.18), we can use Eq. (3.98) to obtain < Ls >=
ρe 1 − ρe
(3.100)
Substituting Eq. (3.99) into Eq. (3.95) we have ρe = ρT −
< Ls > −Ls0 C·T
(3.101)
When ρe is provided by Eq. (3.101), from Eq. (3.100) we have < Ls > −Ls0 C·T =0 < Ls > − < Ls > −Ls0 1 − ρT + C·T ρT −
(3.102)
Equation (3.102) provides an implicit definition of the transition curve < Ls > between E[Ls ] and Ls . Limiting the validity of Eqs. (3.18) and (3.40) to the first quadrant of the plane (ρ; means of Ls ), where they have a physical meaning (See Fig. 3.13), from Eq. (3.102), solved with these restrictions, and since ρT is a current value of traffic intensity (ρ = ρT ), we have 1 2 A +B−A 2
(3.103)
A = (1 − ρ) · C · T + 1 − Ls0
(3.104)
B = 4(Ls0 − ρ · C · T)
(3.105)
< Ls >= with
To give an example of the application of Eq. (3.103), consider the following data (See Fig. 3.8): at the initial time t0 , we obtain Ls0 = 5 veh; starting with t0 , Qe = 1100 veh/h = 0.306 veh/s, C = 980 veh/h = 0.272 veh/s, and, thus, ρ = Qe /C = 1.125. The interval of the system observation is assumed to be T = 10 min = 600 s.
3.3
Time-Dependent Solutions and Waiting Phenomena
93
Fig. 3.13 Domains of time-dependent solutions for the number Ls of users in the system
For < Ls > at t = t0 + T , the quantities A and B are evaluated using Eqs. (3.104) and (3.105) A = (1 − ρ) · C · T + 1 − Ls0 = (1 − 1.125) · 0.272 · 600 + 1 − 5 = −24.4 veh B = 4 · (Ls0 + ρ · C · T) = 4 · (5 + 1.125 · 0.272 · 600) = 754.4 veh and, therefore, having obtained A and B, we get 1 1 2 A +B−A = < Ls >= 2 2
2 ( − 24.4) + 754.4 + 24.4 = 30.57 veh
Following the same procedure that led to Eq. (3.103), it is easy to deduce the transition relationship between the mean of the queue length E[Lc ] under steadystate conditions (See Table 3.3) E[Lc ] =
ρ2 1−ρ
(3.106)
and the mean deterministic value Lc (Eq. (3.38)) Lc (t) = Lc0 + (ρ − 1) · C · T
(3.38)
thus we have for the time-dependent relationship for < Lc > < Lc >=
1 2 D +E−D 2
(3.107)
94
3
Waiting Phenomena at Steady State and Non-steady State Conditions
with D=
(1 − ρ) · (C · T)2 − C · T · Lc0 + 2 · (Lc0 + ρ · C · T) C·T−1 E=
4 · (Lc0 + ρ · C · T)2 C·T−1
(3.108) (3.109)
3.3.2 Time-Dependent Solutions for the Average Time Spent in the System and in the Queue. A Worked Example We refer here to the behaviors of traffic demand Qe and capacity C illustrated in Sect. 3.2.2.1 (See Fig. 3.8); we follow the same procedure illustrated in the same section to obtain the expression of < Ls>. Therefore, we use Eq. 3.17 for the expected value E[ws ] of the average time spent in the system under steady-state conditions 1 1 1 ρ = · 1+ E[ws ] = · C (1 − ρ) C 1−ρ
(3.17)
and we use Eq. 3.49 for the deterministic mean of the same state variable7 ws =
Ls0 + 1 (ρ − 1) · T + C 2
(3.49)
Now, we use Eq. (3.93) or Eq. (3.95) to obtain the transition curve < ws > between E[ws ] and ws ρe = ρT − (ρd − 1)
(3.95)
We recall that (See Fig. 3.12) ρe , ρd , and ρT are values of traffic intensity in correspondence of which it results E[ws ] = ws =<ws>
(3.110)
Therefore, with ρ = ρd and Eq. (3.49) and using Eq. (3.110), we have ρd =
2 Ls0 + 1 · <ws > − +1 T C
(3.111)
7 We recall that w is the mean of the times spent in the system for vehicles that queue up during s the interval T of system observation.
3.3
Time-Dependent Solutions and Waiting Phenomena
95
and with ρ =ρe and Eq. (3.17) and using Eq. (3.110) again <ws > =
1 ρe · 1+ C 1 − ρe
(3.112)
Substituting Eq. (3.111) into Eq. (3.95) we have ρe = ρT +
2 · T
Ls0 + 1 − <ws > C
(3.113)
Using the value of ρe from Eq. (3.113) with Eq. (3.112), we can write ⎛ <ws > −
ρT +
2 T
1 ⎝ · 1+ C 1 − ρT −
·
2 T
Ls0 +1 C
·
⎞
− <ws >
Ls0 +1 C
⎠ = 0
− <ws >
(3.114)
Equation (3.114) implicitly defines the transition curve <ws> between E[ws ] and ws . Limiting the validity of Eqs. (3.17) and (3.49) to the first quadrant of the plane (ρ; means of ws ), where they have a physical meaning, we can solve Eq. (3.114) under these restrictions. Since ρT is a current value of traffic intensity (ρ = ρT ), the result is <ws > =
1 2 J +M−J 2
(3.115)
with T 1 · (1 − ρ) − · (Ls0 + 1) 2 C 4 T 1 M= · · (1 − ρ) + · ρ · T C 2 2 J=
(3.116) (3.117)
Reusing the data of the worked example given in the previous Sect. 3.3.1, we calculate the time-dependent mean of the average time spent in the system for vehicles that joined the queue during the observation interval T = 10 min = 600 s. Then, we use Eqs. (3.116) and (3.117) to calculate J and M 1 · (1 − ρ) − C1 · (Ls0 + 1) = 600 2 · (1 − 1.125) − 0.72 · (5 + 1) = −59.56s $ % $ % 4 1 M = C4 · T2 · (1 − ρ) + 12 · ρ · T = 0.272 · 600 2 · (1 − 1.125) + 2 · 1.125 · 600
J=
T 2
= 4411.76s Finally, from Eq. (3.115), we have
96
3
<ws > =
Waiting Phenomena at Steady State and Non-steady State Conditions
1 1 2 J +M−J = ( − 59.56)2 + 4411.76 + 59.56 = 74.39 s 2 2
Applying the same procedure that led to Eq. (3.115) for the transition relationship between the mean of the waiting time in the queue E[wc ] under steady-state conditions (See the formulas shown in Table 3.3.) 1 E[wc ] = · C
ρ 1−ρ
(3.118)
and the mean deterministic value wc wc =
Lc0 (ρ − 1) · T + C 2
(3.45)
we obtain the time-dependent relationship <wc > <wc > =
1 2
P2 + Q − P
(3.119)
with 1 1 · (1 − ρ) − · (Ls0 − 1) 2 C 2 · Ls0 2·T · ρ+ Q= C C·T
P=
(3.120) (3.121)
With the data of the worked example just developed, we evaluate the average waiting time in the queue <wc >. From Eqs. (3.120) and (3.121) we have P=
1 · (1 − ρ) − C1 · (Ls0 − 1) = 600 2 · (1 − 1.125) − 0.272 · (5 − 1) = −52.21 s $ $ % % 2·Ls0 2·600 2·5 Q = 2·T C · ρ + C·T = 0.272 · 1.125 + 0.272·600 = 5233.56 s T 2
and, therefore, with Eq. (3.119), for <wc > 1 < wc > = 2
1 2 P +Q−P = ( − 52.21)2 + 5233.56 + 52.21 = 70.71 s 2
We note that <ws > − <wc > = 74.39 − 70.71 = 3.68 s This result is in accordance with Eq. (3.4) since <ws > – <wc > = 3.68 s, as indicated by Eq. (3.2), is the value of the mean of service time E[Ts ] = 1/C = 1/0.272 = 3.68 s.
3.3
Time-Dependent Solutions and Waiting Phenomena
97
3.3.3 Generalized Time-Dependent Solutions The previous relationships for the time-dependent mean values of the numbers of users in the system, the queue length and waiting times (< Ls >,
, <ws>, and <wc >) were obtained starting with the expressions of the expected values E[Ls ], E[Lc ], E[ws ], and E[wc ] under steady-state conditions that are valid in the case of Poissonian distributed leg arrivals and exponential service times (See Sect. 3.1). The above-mentioned formulas for the process of arrivals and service times are generally considered to be sufficient to represent most of the situations of practical interest. However, if one wants to have more general expressions than the ones already given by Eqs. (3.103) and (3.115), for example for and <ws >, respectively, we can use Eqs. (3.13) and (3.11) to obtain the expected values E[Ls ] and E[ws ] under steady-state conditions. From these equations, we can start the transition to the deterministic means Ls and ws . E[Ls ] = Qe · s + E[ws ] = s +
Q2e · (s2 + V[s]) 2 · (1 − Qe · s)
Qe · (s2 + V[s]) 2 · (1 − Qe · s)
(3.13) (3.11)
where the mean s = E[Ts ] and the variance V[s] = VAR[Ts ] of the service time are generally given by Eqs. (3.8) and (3.9). In practical applications, s is evaluated, as usual, as s = 1/C, even though this choice may lead to distorted and mutually non-congruent estimations of s and V[s], while, for V[s], one may use the values obtained from traffic data that are available in the literature. Combining Eq. (3.11) with Eq. (3.49) and using the same procedure illustrated in Sects. 3.3.1 and 3.3.2, we have, for example for < ws >, the expression that generalizes Eq. (3.115) 8 using the simplifying assumption that Ls0 = 0. ρ·C T 4·ρ <ws > = s + (s + V[s]) · + ρ − 1 + (ρ − 1)2 + 2 4 C·T
(3.122)
In Fig. 3.14, Eq. (3.122) is shown using these values: s = 6 s; V[s] = 36 s; T = 10 min = 600 s; C = 540 veh/h = 0.15 veh/s. Starting with Eqs. (3.11) and (3.13), relationships similar to Eq. (3.122) can be obtained that generalize Eqs. (3.103), (3.107), and (3.119). Their deduction is left to the keen reader.
8 The behaviors assumed for Q and C are again, as in the other cases analyzed so far, the ones e shown in Fig. 3.8.
98
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Fig. 3.14 Generalized relationship <ws >= < ws (ρ) > for a case with Ls0 = 0
3.3.4 Time-Dependent Solutions for Traffic Peaks Between Two Periods at Steady-State Conditions We refer here to the behaviors of traffic demand and capacity shown in Fig. 3.15. The conditions just before and just after the peak period are the same, and they both show undersaturation Qe0 = Qe1
C0 = C1
ρ0 = ρ1 < 1
(3.123) (3.124)
At the instant t0 , a traffic peak occurs, with Qe > Qe0 and C < C0 , and the entry is oversaturated (ρ > 1). At the end of the peak period for which the duration is T, the entry instantly returns to undersaturation conditions at t = t0 + T (Eq. (3.123)). It is straightforward to note that the behaviors of the cumulative arrival A(t) and departure D(t) values are of the type shown in Fig. 3.10 for this case when Lc0 = Lc1
(3.125)
For the reader’s convenience, the above-mentioned behaviors are also shown in Fig. 3.15. Through the transformation of coordinates technique (Eq. (3.92)) or (Eq. (3.95)), for the case being discussed here that starts with the same steady-state solution (E[wc ], E[ws ]), it is possible to identify, in principle, more than one time-dependent
3.3
Time-Dependent Solutions and Waiting Phenomena
99
Fig. 3.15 Traffic peak between two steady-state conditions
relationship relative to the mean waiting times <wc > and <ws > under a transient condition, according to the deterministic formula on which the deductions are based. In previous Sects. 3.2.2 and 3.2.3, we gave various relationships for wc and ws in the case of a queue at the beginning and at the end of the peak period T, each valid under specific assumptions about the number of vehicles on which we must distribute the total waiting time Wc caused by the traffic peak Qe (only the vehicles that arrived during T or all those affected by the traffic peak). However, by most of the above-mentioned relationships, it is not easy (or possible) to obtain closed-form relationships for <wc> and <ws>.
100
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Thus, among the time-dependent solutions obtainable, the most widely used today is still the one developed by Kimber and Hollis [4] that we will now derive. We operate in terms of increase in the waiting times with respect to the steadystate conditions. If the system instantly passes from a steady-state condition (characterized by a traffic intensity value ρ0 ) to another condition (characterized by a traffic intensity value ρ), the difference E[wc ] between the corresponding expected values of the waiting times E[wc0 ] and E[wc ], if the arrivals are Poissonian and the service times are exponential (See Table 3.3) is 1 E[wc ] = · C
ρ 1−ρ
1 · − C0
ρ0 1 − ρ0
(3.126)
The deterministic increase wc of the average waiting time in the queue with respect to E[wc ], caused by the temporary oversaturation of the entry during T, may be evaluated. This is done by starting with Eq. (3.86) and subtracting from the area w∗∗ c = S(abcdefa), which gives the total waiting time in the queue (see Fig. 3.15), the area S(abgdefa), which has the same value of the cumulative probabilistic waiting time. Thus, we have, with B from Eq. (3.56): 1/2 · (A∗∗ + B) − Lc0 (T + T d ) 1/2 · (A∗∗ + B) − Lc0 (T + T d ) = Qe · T ρ·C·T (3.127) where A∗∗ must be evaluated by substituting L2c1 = L2c0 into Eq. (3.68). In terms of traffic intensity ρ and capacity C before, during, and after the peak, Eq. (3.127), by means of simple transformations, becomes wc =
1 wc = · 2
ρ−1 ρ
(1 − ρ0 ) · C0 + (ρ − 1) · C · ·T (1 − ρ0 ) · C0
(3.128)
To obtain the transition curve <wc > from E[wc ] to wc , we use Eq. (3.92) or Eq. (3.95). ρe = ρT − (ρd − 1)
(3.95)
Even in this case (See Fig. 3.12), we indicate the generic values of traffic intensity with ρe, ρT , and ρd , with the following results E[wc ] = wc = <wc>
(3.129)
Now, to make ρd from Eq. (3.128) explicit after it has been substituted for ρ, we use transformations of variables to simplify the calculation (1 − ρ0 ) · C0 = C − y · ρ
(3.130)
3.3
Time-Dependent Solutions and Waiting Phenomena
101
Thus, using Eqs. (3.130) and (3.129), we obtain ρd =
C · (2 · wc /T) + C − y y · (2 · wc /T) + C − y
(3.131)
that is, since, by Eq. (3.129), in correspondence of ρd , wc = <wc > ρd =
C · (2 · <wc >/T) + C − y y · (2 · <wc >/T) + C − y
(3.132)
With this last expression of ρd inserted into Eq. (3.95) we have ρe = ρT −
C · (2 · <wc >/T) + C − y −1 y · (2 · <wc >/T) + C − y
(3.133)
Now, we specify Eq. (3.126) with ρ = ρe , taking into account Eq. (3.129) 1 <wc > = · C
ρe 1 − ρe
1 · − C0
ρ0 1 − ρ0
(3.134)
If we express ρe in Eq. (3.134) using its mathematical relationship from Eq. (3.133), we finally have the implicit equation of the transition curve <wc > between E[wc ] and wc . ⎫ ⎧ ⎪ ⎪ <w > 1 ⎪ c $ % ⎪ ⎪ ⎪ ⎪ ⎪ y C · ρ− T ⎬ ⎨ ·<w >+ c 1 ρ0 C−y 2 · − <wc> + = 0 (3.135) ⎪ C0 1 − ρ0 1 − ρ + $ y <wc > T % ⎪ ⎪ ⎪ ⎪ ⎪ ·<w >+ ⎪ ⎪ c C−y 2 ⎭ ⎩ Equation (3.135), solved with respect to <wc> with reference to only the first quadrant of the plane (ρ; increases of (·)), since ρT is a current value of traffic intensity (ρT = ρ), leads to <wc > =
1 2 A +B−A 2
(3.136)
with A=
1 1/2 · (1 − ρ) · (C − y) · T + 1/C · [C − y · (1 + ρ)] · + C−y·ρ C0
B=
2·T·
$
1 C
· ρ − (1 − ρ) ·
1 C0
C−y·ρ
·
ρ0 (1−ρ0 )
%
ρ0 1 − ρ0 (3.137)
· (C − y) (3.138)
102
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Finally, the average waiting time in the queue <wc > is obtained by adding the mean waiting time E[wc ] at statistical equilibrium to the increase <wc > 1 · <wc > = <wc > + C0
ρ0 1 − ρ0
(3.139)
From Eqs. (3.139) and (3.4), we obtain the average time spent in the system 1 · <ws > = <wc > + C0
ρ0 1 − ρ0
+ Ts
(3.140)
Equation (3.139), deduced according to Kimber and Hollis, is generally regarded as the conventional average waiting time in the queue (See Sect. 3.2.3). It was obtained starting with a total incremental delay9 represented by the numerator of Eq. (3.127) that, although relative to all the vehicles affected by the traffic peak (vehicles that arrived during the peak period T and the following period T d , which is still affected by the peak effects), was distributed only over the vehicles that arrived during the peak interval (the denominator of Eq. (3.127)). In other words, the peak vehicles that <wc > (Eq. (3.139)) refers to, are given a average waiting time in the queue greater than the one that really pertains to them. In fact, the total waiting time (Eq. (3.127)) includes the waits caused but not endured by peak vehicles. The waiting times that are not endured by the peak vehicles are instead endured by the vehicles arrived during T d , which are still affected by the effects of peak demand. It follows that, for the mean Ts of the service time to insert into Eq. (3.140) to obtain from <wc >, the mean <ws > of the times spent in the system should be use the weighted mean of the service times 1/C and 1/C1 = 1/C0 on the respective times of application. (See illustrations in previous Sects. 3.2.1 and 3.2.2 concerning the calculation of Ts .) A simpler way would be to use only the mean value Ts = 1/C relative to the peak period T [8]. Besides, we may rewrite A (Eq. (3.137)) and B (Eq. (3.138)) in terms of demand (Qe and Qe0 = Qe1 ) and capacity (C and C0 = C1 ). Since ρ = Qe /C and ρ0 = ρ1 = Qe0 /C0 , y (Eq. (3.130)) is equal to y = [C–(1–Qe0 /C0 )·C0 ]/(Qe /C). From the Eq. (3.135), we obtain (flows and capacity veh/h; T in h): <wc > = 1 <ws > = 2
9
1 2 F + G − F + E · 3600 (s) · 2 1 2 F +G−F +E+ · · 3600 (s) C
(3.141) (3.142)
with respect to the cumulative delay derived from the steady-state condition characterized by ρ0 <1.
3.3
Time-Dependent Solutions and Waiting Phenomena
103
with 1 T h F= · · (C − Qe ) · z + α · z − +E C0 − Qe0 2 C 2·T·z Q G= · α · e − (C − Qe ) · E C0 − Qe0 C α · Qe0 E= C0 · (C0 − Qe0 ) h = C − C0 + Qe0 z=1−
h Qe
(3.143) (3.144) (3.145) (3.146) (3.147)
The parameter α was introduced by Kimber and Hollis in addition to the variables of the starting formula (Eq. (3.136)). It was added after the validation (and thus setting) of Eq. (3.136) in order to make a comparison with the results obtained by a different calculation criterion for waiting phenomena under a transient condition. The parameter α was introduced to take into account peak periods T of very short duration (e.g., T = 2–3 min) and the suggested value for α is 2. With α = 2, however, one obtains unrealistic results for values of peak periods that occur more frequently (e.g., with durations in the range of 10–15 min), so, under these circumstances, one must use α =1. Regarding the calculation of the percentiles Ls,p of the number of users in the system Ls during the traffic peak, it implies the determination of the distribution functions FLs (·) of Ls under a transient state. Also, for FLs (·) as was just illustrated for <wc > and <ws >, we can heuristically use the transformation of coordinates technique (Eq. (3.95)), because it is difficult to use the exact results of the mathematical queuing theory under a transient condition. With the above-mentioned criterion applied to the transition, Wu [9] obtained FLs (·) and thus the expression of the percentiles Ls,p of the number of users in the system during the traffic peak. To do so, Wu used the formulation of the distribution function FLs (·) at a statistical equilibrium of Ls to the deterministic law of the same parameter as a function of traffic intensity ρ. The deductive procedure of FLs (·) and, thus, of the percentiles Ls,p , is shown in a detailed and clear way in [10]. We refer to it for the development of the calculations that lead to the following relationship
Ls,p
C·T 8ρ = ρ − 1 + (1 − ρ)2 + − ln γ 4 C·T
(3.148)
where p γ= 1− 100
(3.149)
104
3
Waiting Phenomena at Steady State and Non-steady State Conditions
Thus, for example, the estimation of the 95th percentile of Ls (with γ = 0.05) is
Ls,95
C·T 8ρ = ρ − 1 + (1 − ρ)2 + ·3 4 C·T
(3.150)
while, with γ = 0.01, the 99th percentile of Ls is obtained from the relationship Ls,99
C·T 8ρ = ρ − 1 + (1 − ρ)2 + · 4,6 4 C·T
(3.151)
However, in steady-state conditions, as a first approximation, the double of the average value of Ls,p generally represents an acceptable estimation for the application of a percentile p sufficiently high (p = 90–95) of the number of users in the system, i.e. Ls,p ∼ = 2 E[Ls ]. In the next chapter, we will illustrate some detailed practical applications of the time-dependent solutions presented in this section.
3.3.5 Concluding Remarks Except for Eq. (3.122), the time-dependent relationships described in the previous sections were obtained by assuming that statistical equilibrium solutions, which then allow us to reach deterministic solutions, are relative to Poissonian arrivals and exponential service times. Also, in the case illustrated in Sect. 3.3.4, the assumption was made that statistical equilibrium conditions before and after the peak period are the same (Qe0 = Qe1 ; C0 = C1 ). The plausibility of these assumptions is proved by theoretical research and by the good results in technical practice. More general relationships than the ones deduced starting with the formulation just recalled of Poissonian arrivals and exponential service times (for example, Eq. (3.122)) are described in [4], and the reader should refer to this reference for additional information. However, regardless of the levels of generalization used, the results obtained by the time-dependent formulas, given their heuristic nature, do not completely agree with the exact results obtained through the mathematical queuing theory in the absence of statistical equilibrium. However, the discrepancies for the practical applications of interest are negligible. As we have often stressed in this chapter, time-dependent relationships (See Fig. 3.12) are continuous functions of traffic intensity ρ. This ρ can vary inside its entire domain interval (ρ∈[0;+∞[). This allows us to obtain solutions in the neighborhood of ρ = 1 (where both probabilistic and deterministic approaches give unrealistic results) and to treat the
References
105
case of a traffic peak that is characterized by entry undersaturation during the peak period. In other words, it is possible to examine demand situations that can be defined as peak situations, since the value of Qe in T is greater than Qe0 and Qe1 , which pertain to the intervals before and after T (Qe > Qe0 ; Qe > Qe1 ), respectively, but such that the entry remains constantly undersaturated (Qe < C, i.e ρ < 1) for each t. This type of circumstance will be examined in detail through some worked examples in the next chapter. Finally, it is worth noting that the approximate solutions for studying the transient states of waiting phenomena at intersections are being thoroughly investigated [10]. This proves that this topic is considered to be essential in the field of functional design of intersections.
References 1. Cassandras, C. G., Introduction to Discrete Event Systems, Springer, New York, 1999. 2. Louah, G., Priority Intersections: Modelling, in Concise Encyclopedia of Traffic and Transportation System (Papageorgiu, M., Editor), Pergamon Press, Oxford, 1991. 3. Kleinlock, L., Queuing Systems – Vol. I, John Wiley and Sons, New York, 1975. 4. Kimber, R. M., Hollis, E. M., Traffic Queues and Delays at Road Junctions, TRRL Laboratory Report 909, Crowthorne, Berkshire, 1979. 5. Brilon, W., Delays at Oversaturated Unsignalized Intersections, Based on Reserve Capacities, Transportation Research Record 1484, Washington DC, 1995. 6. Newell, G. F., Queues – Inventories and Maintenance, 3rd edition, John Wiley & Sons, New York, 1976. 7. Morse, P. M., Application of Queuing Theory, 2nd edition, Chapman Hall, London, 1982. 8. Kimber, R. M., Hollis, E. M., Peak-Period Traffic Delays at Road Junctions and Other Bottlenecks, Traffic Engineering and Control (TEC), October 1978. 9. Wu, N., An Approximation for Distribution of Queue Lengths at Unsignalized Intersections, Proceedings of Second International Symposium on Highway Capacity, Vol. 2, ARRB, Sydney, 1994. 10. Brilon, W., Time Dependent Delay at Unsignalized Intersections, Proceedings of 17th International Symposium on Transportation and Traffic Theory (ISTTT17), London, July 2007.
Chapter 4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
With the formulas described in the previous chapter, it is possible to evaluate the waiting times and the number of vehicles in the queue and in the system. This can be done by following the time evolution of traffic demand at legs and by the calculation procedure of capacity given in Chap. 2. According to the level of description of demand variation in time, different computational procedures are used. If demand Qe is considered for successive limited1 period of Tk (generally having the same t value) and is approximated by a continuous function (See Fig. 4.1), the state variables may be performed following step by step the system evolution (See Sect. 4.2). If Qe values are relative to greater2 time periods (or by adding determinations relative to successive periods, See Fig. 4.2), the state variables can be obtained by the procedure of Sect. 3.3.4. The choice among the above-mentioned options depends on various considerations, among which are the availability of more or less detailed measurements (or estimations) of Qe and the aims of the analysis. In the remaining part of this chapter, we will illustrate some important practical applications of the results described in Chap. 3 to roundabouts, starting with the case of a system evolution in which the entries are systematically undersaturated.
4.1 State Evolutions with Undersaturated Entries First, let us consider the case of system evolution among conditions that can be considered as steady-state. We recall that, for technical aims, a steady-state condition is reached for undersaturated entries, if the traffic demand at the intersection is constant for a finite time period T, which is long enough to allow the stabilization of the operating conditions
1
In current technical practice, the values of these intervals are generally in the range of 5−15 min. In current technical practice, the values of these intervals are generally in the range of 30 min to an hour. 2
R. Mauro, Calculation of Roundabouts, DOI 10.1007/978-3-642-04551-6_4, C Springer-Verlag Berlin Heidelberg 2010
107
108
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
Fig. 4.1 Time evolution of traffic demand, capacity, and traffic intensity at an entry approximated by continuous functions
of the roundabout in the neighborhood of constant mean values of the state variables. In addition, the punctual values of state variables must be little dispersed around the mean values E [·]. The transition from one to another of these statistical equilibrium conditions is assumed as instantaneous. In this connection, it is possible to demonstrate [1] that an approximated measurement of the time ϑi , which the system must pass, at a generic entry “i”, from a steady condition, characterized by a traffic intensity ρi0 , to another steady condition characterized by ρi1 , is given by (footnote no. 4 in Sect. 1.1)
4.1
State Evolutions with Undersaturated Entries
109
Fig. 4.2 Time evolution of traffic demand, capacity, and traffic intensity at an entry approximated by step functions
1 1 ϑi = √ √ 2 = √ 2 C · (1 − ρi1 ) ( Ci − Qi ) i
(ρi1 ≤ 1)
(4.1)
If ρi1 <<1, the value of ϑi is small, and the new statistical equilibrium condition is reached rapidly (for practical purposes, instantaneously). If ρi1 tends to 1, ϑi becomes large, and, in this case, transient state conditions must be taken into account during ϑi . In other words, if demand (and/or capacity) varies with respect to the previous state for a period Ti , it is possible to neglect the time-dependent aspects of the waiting phenomenon if
110
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
ρi1 < ρic
(4.2)
with
1 ρic = 1 − √ Ci · Ti
2 (4.3)
If, instead, ρi1 > ρic , in the analysis of the system, the results of the statistical equilibrium are not usable, but time-dependent formulas are used. When the roundabout evolves through conditions that can be considered as steady-state conditions, the determinations of the state variables as a function of capacity Ci and of traffic intensity ρi = Qei /Ci are performed with the formulas shown in Table 3.3 for Poissonian arrivals and exponential service times. It is worth noting that the entering demand flows must be increased by the number of vehicles waiting at the end of the previous time segment Tk−1 for the calculation of capacity by assuming that capacity is constant at intervals in the various successive intervals Tk of subdivision of T. (k) In other words, if the interval Tk demand at the generic leg “i” is equal to Qei (k−1) and several vehicles Lsi are already at the entry, the flow entering the circulatory (k) (k) (k) roadway during Tk , Qei ∗ , with Qei and Qei ∗ expressed in pcu/h, is equal to (k)*
Qei
60 (k) Tk (k−1) (k) (k−1) 60 = Qei · = Qei + Lsi · + Lsi · 60 Tk Tk (k−1)
where the number of users in the system Lsi
(4.4)
is 3
(k−1)
= E[Lsi ](k−1) at steady-state conditions
(4.5)
(k−1)
= < Lsi > (k−1) at transient conditions
(4.6)
Lsi
Lsi
If we record zero values of traffic demand at a generic entry “i” in Tk , with Eq. (4.4) it is possible to take into account a flow entering the circulatory roadway that (k−1) vehicles in the system that queued up during the interval Tk–1 . consists of Lsi From Eq. (4.4), it is straightforward to deduce that, if the interval Tk is suffi(k−1) ciently long, the contribution of the waiting vehicles Lsi to the entering flow is negligible, and it becomes null when T is infinite. Now, it is worth noting that, for calculation aims, in all the relations relative to the waiting phenomenon described in Chap. 3, traffic demand and capacity must be expressed in vehicles (veh) per time, generally veh/h or veh/s.
3 We recall that, with E[·], we indicate the expected value and with < · >, we indicate the timedependent mean (Sect. 3.3).
4.1
State Evolutions with Undersaturated Entries
111
In the capacity formulas (See Chap. 2), demand and capacity are measured in passenger car units (pcu) per time, i.e., generally as pcu/h or pcu/s. Therefore, the values of traffic demand and the values obtained by capacity calculations must be converted from one type of measurement to the other in order to be used to determine waiting times, queue length, and the number of vehicles in the system. Capacity and traffic demand values Qe and C (in veh/time units) are obtained by multiplying the determinations of Qe and C by the self-evident factor f if Qe and C are given in pcu/time units f=
1 (1 − Pp − Pcm ) + αp · Pp + αcm · Pcm
(4.7)
where Pp and Pcm are the rates of heavy vehicles (coaches included) and bicycles and motorbikes, respectively, present in the flow Qe . The symbols αp and αcm are used to indicate the coefficients of equivalence of the above-mentioned vehicles in passenger cars (in general, see Sect. 2.1, αp = 2 and αcm = 0.5). (·) On the other hand, the determination of Lsi that is expressed in vehicle units (·) (·) (veh), in order to be added in Eq. (4.4) to the quantity Qei · T· /60 (where Qei is in pcu/h) must be converted, using the factor f provided by Eq. (4.7), into passenger car units (pcu). To calculate traffic intensity ρ, we use one or the other vehicular volume measure, thus obtaining the same numerical result. To better explain the above-mentioned points, we will give some calculation examples in the following sections.
4.1.1 First Worked Example Consider a four-legged roundabout with a single-lane circulatory roadway and single-lane entries. Traffic demand (given in the form shown in Sect. 1.1.1) evolves instantaneously through three stages. To simplify the procedure, we assume that the percentages of vehicles different from passenger cars are completely negligible, so that the values of demand and capacity expressed in pcu/h coincide with those expressed in veh/h. State 0 (from t = 0 to t = t1 ) ⎡
(0) (0) [Qei ] = 680 600 731 550 PO/D
0 ⎢ 0.35 ≡⎢ ⎣ 0.15 0.40
0.40 0 0.30 0.40
0.40 0.50 0 0.20
⎤ 0.20 0.15 ⎥ ⎥ 0.55 ⎦ 0
(4.8)
112
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
⎡
(0)
MO/D
0 ⎢ 210 ≡⎢ ⎣ 110 220
272 0 219 220
272 300 0 110
⎤ 136 90 ⎥ ⎥ 402 ⎦ 0
(4.9)
State 1 (from t = t1 to t = t2 for a duration T1 = 20 min) ⎡
0 0.25 0.36 ⎢ 0.29 (1) 0 0.37 (1) ⎢ [Qei ] = 350 280 404 309 PO/D ≡ ⎣ 0.33 0.29 0 0.31 0.35 0.34 ⎡ ⎤ 0 88 126 136 ⎢ 81 0 104 95 ⎥ (1) ⎥ MO/D ≡ ⎢ ⎣ 132 116 0 156 ⎦ 96 108 105 0
⎤ 0.39 0.34 ⎥ ⎥ 0.38 ⎦ 0
(4.10)
(4.11)
State 2 (from t = t2 for a duration T2 = 30 min) ⎡
0 0,35 0,35 ⎢ 0,30 0 0,35 (2) (2) [Qei ] = 455 460 433 420 PO/D ≡ ⎢ ⎣ 0,15 0,30 0 0,20 0,40 0,40 ⎡ ⎤ 0 159 159 137 ⎢ 138 0 161 161 ⎥ (2) ⎥ MO/D ≡ ⎢ ⎣ 65 130 0 238 ⎦ 84 168 168 0
⎤ 0,30 0,35 ⎥ ⎥ 0,55 ⎦ 0
(4.12)
(4.13)
To calculate capacity, we use the German formula by Brilon-Wu (Eq. 2.12). State 0 By applying Eq. (1.12) from Chap. 1 to the elements of matrix (4.9), we obtain (0) the circulating flows [Qci ] (0) (0) (0) (0) [Qci ] = [ Q(0) ] = [ 549 518 436 539 ] (pcu/h) c1 Qc2 Qc3 Qc4
(4.14)
and, similarly, the capacities are obtained by applying Eq. (2.12) of Chap. 2 (0) (0) (0) (0) [Ci ] = [ C(0) ] = [ 776 800 866 784 ] (pcu/h) 1 C2 C3 C4
For the traffic intensities (degrees of saturation) ρi
(0)
= Qei (0) /Ci
(0) (0) (0) (0) [ρi ] = [ρ(0) ] = [ 0.88 0.75 0.84 0.70 ] 1 ρ2 ρ3 ρ4
(4.15) (0) ,
we have (4.16)
For each entry lane, the capacities and traffic intensities are known, so, using the relationships shown in Table 3.3 of Chap. 3, we calculate the averages of the times spent in the system E[wsi ](0) and the number of the vehicles in the system E[Lsi ](0)
4.1
State Evolutions with Undersaturated Entries
113
[E[wsi ](0) ] = [E[ws1 ](0) E[ws2 ](0) E[ws3 ](0) E[ws4 ](0) ] = = [ 38.66 17.99 25.98 15.31 ] (s) [E[Lsi ](0) ] = [E[Ls1 ](0) E[Ls2 ](0) E[Ls3 ](0) E[Ls4 ](0) ] = = [ 7.33 3.00 5.25 2.33 ] (veh)
(4.17) (4.18)
For the vector of the percentiles, we have [Lsi,p ](0) ∼ = 2 [E[Lsi ](0) ], (p = 90–95, see the end of Sect. 3.3.4) and, therefore [Lsi,p ](0) = [ 14.66 6.00 10.50 4.66 ] (veh)
(4.19)
State 1 (1) To calculate the circulating flows [Qci ], we take into account the number of vehi(0) cles [E[Lsi ] ] waiting at legs calculated for the previous condition (vector (4.18)) that enter the circulatory roadway during the interval T1 = 20 min and are therefore (1) added to those that form demand flow [Qei ]. (1) In conclusion, [Qei ] (Eq. (4.10)) and [E[Lsi ](0) ] (Eq. (4.18)) yield (Eq. (4.4)) the volumes (expressed in pcu/h)4 Qe1 ∗ = 350 + 7.33 ·
60 20
= 372
Qe2 ∗ = 280 + 3.00 ·
60 20
= 289
Qe3 ∗ = 404 + 5.25 ·
60 20
= 420
Qe4 ∗ = 309 + 2.33 ·
60 20
= 316
(1) (1) (1) (1)
(4.20)
that is (1) [Qei ∗ ] = 372 289 420 316 (pcu/h)
(4.21)
With the elements of vector (4.21) and matrix (4.10), Eq. (1.12) of Chap. 1 yields the following circle flows (1) [Qci ∗ ] = 340 386 327 344 (pcu/h)
(4.22)
With these values, we have the determinations of entry capacities during T1 = 20 min from Eq. (2.12) of Chap. 2
4
In this case, we recall that traffic demand consists of only passenger cars, and, therefore, it is not necessary to convert the number of waiting vehicles into passenger car units.
114
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
(1) [Ci ] = 945 909 956 942 (pcu/h)
(4.23)
With vectors (4.10) and (4.23), we calculate the traffic intensity vector ρi Qei (1) /Ci (1) (1)
(1) (1) (1) [ρi ] = [ρ(1) ] = [ 0.37 0.31 0.42 0.33 ] 1 ρ2 ρ3 ρ4
(1)
=
(4.24)
From Eq. (4.1), we obtain the values of each time interval ϑi (1) using the capacities and traffic intensities described in Eqs. (4.23) and (4.24), respectively. (1)
ϑ1 = (1)
ϑ2 = (1)
ϑ3 = (1)
ϑ4 =
1 (1) (1) C1 ·(1− ρ1 )2 1 (1)
(1)
C2 ·(1− ρ2 )2 1 (1) C3 ·(1−
(1) ρ3 )2
1 (1) (1) C4 ·(1− ρ4 )2
=
1 √ (945/3600)·(1− 0.37)2
= 24.8 s
=
1 √ (907/3600)·(1− 0.31)2
= 20.2 s
=
1 √ (956/3600)·(1− 0.42)2
= 30.4 s
=
1 √ (942/3600)·(1− 0.33)2
= 21.1 s
(4.25)
(1)
All the values of ϑi obtained are much smaller than the observation interval T1 = 20 min = 1200 s. The assumption of instantaneous transition from stage 0 to stage 1 that we made at the beginning of this worked example is, therefore, plausible. In other words, the interval T1 = 20 min is sufficiently wide to allow the roundabout to rapidly reach steady-state conditions and preserve them in stage 1. The averages of the times spent in the system (in s) E[wsi ](1) and the number of vehicles in the system E[Lsi ](1) are obtained by using the relationships shown in Table 3.3 of Chap. 3 (Poissonian arrivals and exponential service times) and the values of Ci and ρi described in Eqs. (4.23) and (4.24), respectively: [E[wsi ](1) ] = [E[ws1 ](1) E[ws2 ](1) E[ws3 ](1) E[ws4 ](1) ] = = [ 6.05 5.75 6.50 5.71 ] (s) [E[Lsi ](1) ] = [E[Ls1 ](1) E[Ls2 ](1) E[Ls3 ](1) E[Ls4 ](1) ] = = [ 0.59 0.45 0.72 0.49 ] (veh)
(4.26)
(4.27)
For the percentiles, with p = 90 – 95, we have [Lsi,p ](1) ∼ = 2 [E[Lsi ](1) ] and, therefore, it is [Lsi,p ](1) = [ 1.18 0.90 1.44 0.98 ] (veh)
(4.28)
4.1
State Evolutions with Undersaturated Entries
115
State 2 (2) To calculate the circulating flows [Qei ] to be introduced in the capacity formula, we must now take into account the number of vehicles [E[Lsi ](1) ] in the system (waiting at legs) determined for the previous condition (vector (4.27)) that enter the circulatory roadway during the interval T2 = 30 min, and these vehicles must, (2) therefore, be added to the vehicles that form the demand flow [Qei ]. (2) In conclusion, [Qei ] (Eq. (4.12)) and [E[Lsi ](1) ] (Eq. (4.27)) yield (Eq. (4.4)) the volumes (expressed in pcu/h) Qe1 ∗ = 455 + 0.59 ·
60 30
= 456
Qe2 ∗ = 460 + 0.45 ·
60 30
= 461
Qe3 ∗ = 433 + 0.72 ·
60 30
= 434
Qe4 ∗ = 420 + 0.49 ·
60 30
= 421
(2) (2) (2) (2)
(4.29)
that is (2) [Qei ∗ ] = 456 461 434 421 (pcu/h)
(4.30)
[Qei ∗ ] actually coincides with the vector [Qei ] (vector (4.12)). With the elements of vector (4.30) and matrix (4.12), Eq. (1.12) of Chap. 1 yields the circle flows that follows (2)
(2)
(2) [Qci ∗ ] = 466 464 436 334 (pcu/h)
(4.31)
With these values, we can calculate the entry capacity determinations during T2 = 30 min using Eq. (2.12) of Chap. 2. (2) [Ci ] = 842 843 866 950 (pcu/h) The traffic intensity vector ρi (2)
(2)
= Qei (2) /Ci
(2)
(4.32)
is
(2) (2) (2) ] = [ 0.54 0.55 0.50 0.44 ] [ρi ] = [ρ(2) 1 ρ2 ρ3 ρ4
(4.33)
With capacity (4.32) and traffic intensities (4.33), from Eq. (4.1), we have the (2) values for each time interval ϑi
116
4 (2)
ϑ1 = (2)
ϑ2 = (2)
ϑ3 = (2)
ϑ4 =
Calculation of Waiting Times, Queue Lengths, and Levels of Service 1 (2) (2) C1 ·(1 - ρ1 )2 1 (2)
(2)
C2 ·(1 - ρ2 )2 1 (2) C3 ·(1 (2)
1
C4 ·(1 -
(2) ρ3 )2 (2)
ρ4 )2
=
1 √ (842/3600)·(1 - 0.54)2
= 60.8 s
=
1 √ (843/3600)·(1 - 0.55)2
= 63.9 s
=
1 √ (866/3600)·(1 - 0.50)2
= 48.5 s
=
1 √ (950/3600)·(1 - 0.44)2
= 33.4 s
(4.34)
All the values of ϑi (2) obtained are much smaller than the observation interval T2 = 30 min = 1800 s. The assumption of instantaneous transition from stage 2 to stage 3 that we made at the beginning of this worked example is, therefore, plausible. In other words, the interval T2 = 30 min is sufficiently wide to allow the roundabout to rapidly reach steady-state conditions and preserve them in stage 2. The averages of the times spent in the system (in s) E[wsi ](2) and the number of vehicles in the system E[Lsi ](2) can be obtained from the relationships shown in Table 3.3 of Chap. 3 using the values of capacity and traffic intensity described in Eqs. (4.32) and (4.33), respectively
[E[wsi ](2) ] = [E[ws1 ](2) E[ws2 ](2) E[ws3 ](2) E[ws4 ](2) ] = = [ 9.30 9.49 8.31 6.77 ] (s)
(4.35)
[E[Lsi ](2) ] = [E[Ls1 ](2) E[Ls2 ](2) E[Ls3 ](2) E[Ls4 ](2) ] = = [ 1.17 1.22 1.00 0.79 ] (veh)
(4.36)
For the percentiles, with p = 90 – 95, we have [Lsi,p ](2) ∼ = 2 [E[Lsi ](2) ], and, therefore, it is
[Lsi,p ](2) = [ 2.34 2.44 2.00 1.58 ] (veh)
(4.37)
4.1.2 Second Worked Example Assume that the roundabout described in the previous example evolves again through three states, with the first and the third associated with traffic demand (pcu/h) given by
4.1
State Evolutions with Undersaturated Entries
117
(0) (2) [Qei ] = [Qei ] = 590 540 500 530 ⎡
(0)
(2)
PO/D ≡ PO/D
0 ⎢ 0.35 ≡⎢ ⎣ 0.15 0.40 ⎡
(0)
(2)
MO/D ≡ MO/D
0 ⎢ 210 ⎢ ≡⎣ 110 220
0.40 0 0.30 0.40
0.40 0.50 0 0.20
272 0 219 220
272 300 0 110
⎤ 0.20 0.15 ⎥ ⎥ 0.55 ⎦ 0 ⎤ 136 90 ⎥ ⎥ 402 ⎦ 0
This demand is applied for a duration that can be considered, for practical aims, infinite. The second state, the duration of which is T1 = 10 min, is connected to the matrix (1) (1) PO/D (4.10) and the vector [Qei ] (1) [Qei ] = [ 638 590 660 680 ] (pcu/h)
(4.38)
(1)
that is to matrix MO/D ⎡
(1)
MO/D
0 ⎢ 171 ⎢ ≡⎣ 218 211
159 0 191 238
230 218 0 231
⎤ 249 201 ⎥ ⎥ (pcu/h) 251 ⎦ 0
(4.39)
Also in this example, to simplify the procedure, we assume that, in the flows under examination, the percentages of vehicles different from passenger cars are completely negligible, so that the values expressed in pcu/h and pcu coincide with those expressed in veh/h and veh. Again, we use the capacity formula developed by Brilon-Wu (See Sect. 2.1.5). State 0 By the same calculation criteria used for the homologous condition of the worked example described in Sect. 4.1.1, we have (0) [Qci ] = [ 468 460 388 414 ] (puc/h) (0) [Ci ] = [ 840 847 905 884 ] (puc/h) (0) [ρi ] = [ 0.70 0.64 0.55 0.60 ] [E[wsi ](0) ] = [ 14.28 11.81 8.84 10.18 ] (s)
[E[Lsi ](0) ] = [ 2.33 1.78 1.22 1.50 ] (veh) [Lsi,p ](0) = [ 4.66 3.56 2.44 3.00 ] (veh)
118
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
State 1 (1) To calculate the circulating flows [Qei ], we take into account the number of vehicles [E[Lsi ](0) ], determined for the previous condition, that enter the circulatory roadway during the interval T1 = 10 min and that must, therefore, be added to the (1) vehicles that form the demand flow [Qei ]. (1) Finally, [Qei ] (Eq. (4.38)) and [E[Lsi ](0) ] yield, on the basis of Eqs. (4.4) and (4.5), the volumes (expressed in pcu/h) (1) Qe1 ∗ = 638 · (1) Qe2 ∗ = 590 · (1) Qe3 ∗ = 660 · (1) Qe4 ∗ = 680 ·
10 60 10 60 10 60 10 60
+ 2.33 · + 1.78 · + 1.22 · + 1.50 ·
60 10 60 10 60 10 60 10
= 652 = 601 = 667
(4.40)
= 689
that is (1) [Qei ∗ ] = 652 601 667 689 (pcu/h)
(4.41)
(1)
With the elements of vector (4.41) and matrix PO/D (matrix 4.10), Eq. (1.12) of Chap. 1 yields the circle flows that follow (1) [Qei ∗ ] = 669 723 633 589 (pcu/h)
(4.42)
With these values, we can calculate entry capacities during T1 = 10 min from Eq. (2.12) of Chap. 2. (1) [Ci ] = 683 643 711 745 (pcu/h)
With vectors (4.38) and (4.43) we calculate the traffic intensity vector ρi Qei (1) /Ci (1) (1)
(1) (1) (1) [ρi ] = [ρ(1) ] = [ 0.93 0.92 0.93 0.91 ] 1 ρ2 ρ3 ρ4
(4.43)
(1)
=
(4.44)
With the capacities and traffic intenities described in Eqs. (4.43) and (4.44), respectively, from Eq. (4.1) we have the values for each time interval ϑi (1)
4.1
State Evolutions with Undersaturated Entries (1)
ϑ1 = (1)
ϑ2 = (1)
ϑ3 = (1)
ϑ4 =
1 (1) (1) C1 ·(1− ρ1 )2 1 (1)
(1)
C2 ·(1− ρ2 )2 1 (1) C3 ·(1−
(1) ρ3 )2
1 (1) (1) C4 ·(1− ρ4 )2
119
=
1 √ (683/3600)·(1− 0.93)2
= 4148s = 69.1 min
=
1 √ (643/3600)·(1− 0.92)2
= 3359s = 56.0 min
=
1 √ (711/3600)·(1− 0.93)2
= 3988s = 66.5 min
=
1 √ (745/3600)·(1− 0.91)2
= 2279s = 38.0 min
(4.45)
(1)
All the values of ϑi obtained are much greater than the observation interval T1 = 10 min = 600 s. The assumption of instantaneous transition from stage 0 to stage 1 is, therefore, not plausible. In other words, the interval T1 = 10 min is not sufficiently wide to allow the roundabout to reach steady-state conditions and preserve them in stage 1. Therefore, the averages of the times spent and the number of vehicles in the system cannot be obtained from the relationships that are valid for the steady-state conditions. Instead, they must be calculated starting with Eq. (3.142) and with the specifications of Eqs. (3.143)–(3.147), all of which are contained in Chap. 3 (flows and capacities in pcu/h; time T in h). <wsi > =
1 · 2
1 2 Fi + Gi − Fi + Ei + (1) Ci
· 3600 (s)
(4.46)
where the subscript “i” refers to the factors of the second member relative to the generic entry “i:” ! " T1 hi (1) (1) · · (Ci − Qei ) · zi + zi − (1) + Ei Fi = (0) (0) 2 Ci − Qei Ci (1) Qei 2 · T1 · zi (1) (1) Gi = (0) · − (Ci − Qei ) · Ei (0) (1) Ci − Qei Ci 1
(4.47) (4.48)
(0)
Ei =
Qei (0)
(0)
(0)
(1)
(0)
(0)
Ci · (Ci − Qei )
hi = Ci − Ci + Qei zi = 1 −
hi (1)
(4.49) (4.50) (4.51)
Qei
Substituting the values of traffic demand and capacity calculated before for state 0 and state 1 into Eqs. (4.47)–(4.51) for entry 1, we have
120
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service (1)
(0)
(0)
h1 = C1 − C1 + Qe1 = 683 − 840 + 590 = 433 z1 = 1 −
h1 (1) Qe1
=1−
(0)
Qe1
E1 =
F1 =
(0) C1
(0) · (C1
=
590 = 0.0028095 840 · (840 − 590)
1 (0)
(0) − Qe1 )
433 = 0.3213166 638
(0)
C1 − Qe1
! " T1 h1 (1) (1) · · (C1 − Qe1 ) · z1 + z1 − (1) + E1 = 2 C1
433 10 1 · · (683 − 638) · 0.3213166 + 0.3213166 − = 840 − 590 2 683 + 0.0028095 = 0.0063787 G1 =
=
2 · T1 · z1 (0) C1
(0) − Qe1
·
(1)
Qe1
(1) C1
(1)
(1)
−(C1 − Qe1 ) · E1 =
638 2 · 10 · 0.3213166 · − (683 − 638) · 0.0028095 = 0.0003460 840 − 590 683
and, therefore, <ws1
>(1)
=
1 · 2
1 F21 + G1 − F1 + E1 + (1) C1
· 3600 =
1 1 2 (0.0063787) +0.0003460−0.0063787 +0.0028095+ · · = 2 683
3600 = 39.3 s Repeating the calculation just performed for entry 1, for the remaining three entries we have <ws2 >(1) = 36.5 s <ws3 >(1) = 34.3 s <ws4 >(1) = 31.9 s To determine the 95th percentile of Lsi we use Eq. (3.150) from Chap. 3, and, thus, we have
4.2
State Evolution with Saturated or Oversaturated Entries
121
⎤ . (1) . 8 · ρ ·T (1) (1) Ls1.95 = · ⎣ρ1 − 1 + /(1 − ρ1 )2 + (1) 1 · 3⎦ = 4 C1 · T 683 · 0.167 8 · 0.93 = · 0.93 − 1 + (1 − 0.93)2 + · 3 = 10.8 veh 4 683 · 0.167 (1) C1
⎡
Ls2.95 = 10.2 veh Ls3,95 = 10.9 veh Ls2,95 = 10.5 veh Applying in any case, to stage 1, the relationship at steady-state conditions (See Table 3.3 of Chap. 3,) we have the following values
[Lsi,95
[E[wsi ](1) ] = [ 75.3 70.0 72.4 53.7 ] = [ 26.6 23.0 26.6 20.2 ] (veh)
](1)
by which we overestimate the determinations of the state variables under examination. State 2 (2) To calculate the circulating flows [Qci ] to be introduced into the capacity formula, we must take into account the number of vehicles < Lsi > (1) in the system (waiting at legs) at the end of the previous period T1 . However, since the value of (2) T2 is in this example assumed to be infinite, the entering flows [Qei ∗ ] coincide with the demand flows (Eq. (4.4)). (0) (2) In addition, having assumed that traffic demand is equal to [Qei ] = [Qei ] and (0) (2) MO/D = MO/D , it follows that, under this condition, the determinations of [E[wsi ](2) ] and [E[Lsi ](2) ] coincide with those of state 0. In the end, we have [E[wsi ](2) ] = [ 14.28 11.81 8.84 10.18 ] (s) [E[Lsi ](2) ] = [ 2.33 1.78 1.22 1.50 ] (veh) [Lsi,95 ](2) = [ 4.66 3.56 2.44 3.00 ] (veh)
4.2 State Evolution with Saturated or Oversaturated Entries In the case of evolution of the system between conditions with saturated or oversaturated entries, we use time-dependent formulas since there are no steady-state conditions. For each entry, the behavior of traffic demand with time is known (assigned, for example, as in the form in Sect. 1.1).
122
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
The beginning and the end of observation interval T are selected in correspondence with the system under steady-state conditions. T is divided into subintervals Tk , during which Qei (t) is constant (See Fig. 4.1). The calculation of the state variables of the system is performed with the following procedure: – we select a capacity formula for the roundabout under examination; – for the initial steady-state condition, we determine, with the criteria described in Sect. 1.1, the circle flows, the exiting flows, and the disturbing volumes to be introduced into the capacity formula selected in correspondence with each leg; – we calculate the first capacity value for each entry “i”; – then, for each entry, we use the relationships described in Sect. 3.1 to determine the average time spent [E[wsi ](0) ] and the number of vehicles in the system [E[Lsi ](0) ] (and/or the waiting time in the queue and the queue length if we also wish to use these state variables) at statistical equilibrium; – at each successive calculation step k (with the respective interval Tk ), we determine for each entry: (k)∗
• the demand value Qei , on the basis of Eq. (4.4), as a function of the value (k) Qei of traffic demand during Tk and of the number of vehicles in the system at the end of the previous period Tk–1 ; (k)* • with the Qei relative to each entry, the circle flows, the exiting flows, and the conflicting flows to be introduced into the selected capacity formula; (k) • the capacities Ci at each entry (in case of saturation or oversaturation of one or more of the entries using the procedure illustrated in Sect. 2.5); (k) (k) (k) • the traffic intensities (degrees of saturation) ρi = Qei /Ci ; • the number of users in the system (k) at the end of the interval Tk ((Eq. (3.103) of Chap. 3) using, for each entry “i” during the interval Tk , Eqs. (3.104) and (3.105) of Chap. 3 with: (k)
(k)
ρ = ρi ; C = Ci ; T = Tk ;
Ls0 = (k−1)
thus, obtaining at step k (k) =
1 2
(k) (k) (k) (Ai )2 + Bi − Ai
(4.52)
with (k)
Ai
(k)
(k)
= (1 − ρi ) · Ci · Tk + 1 − (k−1) )
(4.53)
4.2
State Evolution with Saturated or Oversaturated Entries (k)
123 (k)
Bi = 4((k−1) +ρ i · Ci · Tk )
(4.54)
We repeat that the value Ls0 of Eq. (3.104) of Chap. 3 evidently coincides with the number of users in the system (k - 1) at the end of interval Tk–1 previous to the one considered in the calculation (at the beginning of the calculation procedure we have k = 1 and Ls0 = E[Lsi ](0) ); • the average time spent in the system <wsi >(k) during Tk , with Eq. (3.115) of Chap. 3. Equation (3.115) of Chap. 3 is specified for each entry “i” putting into Eqs. (3.116) and (3.117) of Chap. 3: (k)
(k)
ρ = ρi ; C = Ci ; T = Tk ; Ls0 = (k−1) thus, we obtain at step k: < wsi >(k) =
1 2
(k) (k) (k) (Ji )2 + Mi − Ji
(4.55)
with (k)
Ji
=
(k)
Mi
Tk 1 (k) · (1 − ρi ) - (k) · (< Lsi >(k−1) + 1) 2 Ci
=
4 (k)
Ci
·
Tk 1 (k) (k) · (1 − ρi ) + · ρi · Tk 2 2
(4.56)
(4.57)
The procedure ends when the entire system observation interval has been covered. Evidently, if we are interested in the determination of the queue lengths and waiting times in the queue, we must also specify, with the same positions used for Eqs. (3.103)–(3.105) and for Eqs. (3.115)–(3.117), Eqs. (3.107)–(3.109), and Eqs. (3.119)–(3.121) of Chap. 3. So, we must use Eqs. (3.107) and (3.119) from Chap. 3. To better explain the points illustrated so far, we will now give two calculation examples.
4.2.1 First Worked Example At the beginning of the computational process, the roundabout that is assumed to have four legs (i = 1, 2, 3, 4) is at steady-state conditions.
124
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
Traffic demand has the following percentage composition: heavy vehicles (coaches included) account for 6.5%, and bicycles and motorcycles account for 5% of the total volume Qei . This composition remains the same on all legs of the system during the entire observation interval T. For the transformation from pcu/h and pcu into veh/h and veh, respectively, the coefficient f given by Eq. (4.7) is f=
1 = 0.962 (1 − 0.065 − 0.05) + 2 · 0.065 + 0.5 · 0.05
where we have used the values αp = 2 and αcm = 0.5, respectively, for the equivalence in passenger cars. In addition, the matrix PO/D of the traffic percentages is also assumed to be invariant during the entire T. The variability of demand in time is thus attributable only to (k) the behavior of flows Qei during the sub-intervals Tk (See Fig. 4.3). To give an example, we follow the evolution process relative to entry 2 (The procedure can be repeated for the other entries). (0) (0) From the calculation of disturbing flows Qdi on the basis of traffic demand (Qei ; PO/D ), with the selected capacity formula we have, for entry 2, the capacity value (0) C2 = 1055 pcu/h = 1055 · 0.962 = 1015 veh/h. (0) (0) As Qei = 895 pcu/h (See Fig. 4.3), for the traffic intensity ρ2 = 895/ 1055 = 0.848 and, therefore, for the number of vehicles E[Ls2 ](0) and the time spent E[ws2 ](0) in the system (See Table 3.3 of Chap. 3), assuming Poissonian arrivals and exponential service times, we have: E[ws2 ](0) = 1/[(1015/3600) · (1 − 0.848)] = 23.3 s E[Ls2 ](0) = 0.848/(1 − 0.848) = 5.6 veh (0)
(0)
Similarly, since we know Qei and have determined Ci for the other three entries, we have the initial values [E[wsi ](0) ] and E[Lsi ](0) ]. Demand evolution is given for intervals of Tk = 10 min. In step 1, we calculate (Eq. (4.4)), for each entry, the increase in demand caused by the number of vehicles waiting in the system relative to the previous state; to do so, the [E[Lsi ](0) ] must be converted from veh to pcu, once again by using the coefficient f = 0.962. (1) Thus, for entry 2, since Qe2 = 1034 pcu/h = 1034 · 0.962 = 1000 veh/h and (0) E[Ls2 ] = 5.6/0.962 = 5.82 pcu/h, we have Qe2 ∗ = 1034 + 5.82 · (1)
60 = 1069pcu/h 10
Since we know the demand [Qei ∗ ] we determine, for each entry, the disturbing (1) flows and the capacity values [Ci ] relative to calculation step 1, i.e., for interval T1 . (1)
4.2
State Evolution with Saturated or Oversaturated Entries
Fig. 4.3 Time evolution of traffic demand, capacity, traffic intensity, and state variables for entry 2
125
126
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
(1)
(1)
For C2 , our calculation yields C2 = 998 pcu/h. The entry is oversaturated, and, therefore, we used the computational procedure described in Sect. 2.5 for the determination of capacity. (1) To determine <ws2 > (1) and (1) , we convert C2 from pcu/h to veh/h. Thus, we have (1)
C2 = 998 · 0.962 = 960 veh/h and we evaluate traffic intensity during the period T1 that is equal to (1)
(1)
(1)
ρ2 = Qe2 /C2 = 1000/960 = 1.04 Putting these values into Eqs. (4.53) and (4.54), we have (1)
(1)
(1)
A2 = (1 − ρ2 ) · C2 · T1 + 1− < Ls2 > (0) = (1 − 1.04) · 960/3600 · 600 + 1 − 5.6 = −11.00 (1)
(1)
(1)
B2 = 4 · ( < Ls2 >(0) +ρ2 · C2 · T1 ) = 4 · (5.6 + 1.04 · 960/3600 · 600) = 688.00 and, therefore, with Eq. (4.52), we have
(1)
1 = · 2
(1) (A2 )2
(1) + B2
(1) − A2
=
1 ( − 11.00)2 + 688.00 + 11.00 = 19.7 veh · 2 By specifying Eqs. (4.56) and (4.57), we have =
(1)
J2 = =
(1)
M2 = =
T1 2
(1)
· (1 − ρ2 ) −
1 (1) C2
· ( < Ls2 > (0) + 1) =
600 1 · (1 − 1.04) − · (5.6 + 1) = −36.75 2 960/3600 $
%
(1) (1) T1 4 1 (1) · 2 · (1−ρ2 ) + 2 · ρ2 · T1 = C2 $ % 4 600 1 · · (1 − 1.04) + · 1.04 · 600 960/3600 2 2
= 4500.00
and, therefore, from Eq. (4.55), we have
(1) <ws2 >
=
= 12 ·
( − 36.75)2 + 4500.00 + 36.75 = 56.6 s
1 2
·
(1) (J2 )2
(1) + M2
(1) − J2
=
4.2
State Evolution with Saturated or Oversaturated Entries
127
In step 2, we calculate (Eq. (4.4)), for each entry, the increase in demand caused by the number of vehicles in the system relative to the previous state; to do so, the (1) must be converted from veh to pcu, once again using the coefficient f = 0.962. (2) Therefore, for entry 2, since Qe2 = 1060 pcu/h = 1000 · 0.962 = 1020 veh/h and (1) = 19.7/0.962 = 20.48 pcu/h, we have
(2)*
Qe2 = 1060 + 20.48 ·
60 = 1183 pcu/h 10
Since we know the demand [Qei ∗ ], we determine, for each entry, the disturbing (2) flows and then the capacity values [Ci ] relative to calculation step 2, that is, for the interval T2 . (2) (2) For C2 , our calculation yields C2 = 982 pcu/h (the entry is oversaturated, and, thus, for the calculation of capacity, we used the computational procedure described in Sect. 2.5). (2) To determine <ws2 >(2) and < Ls2 > (2) , we convert C2 from pcu/h to veh/h. Thus, we have (2)
(2)
C2 = 982 · 0.962 = 945 veh/h and we evaluate traffic intensity during period T2 , which is equal to (2)
(2)
(2)
ρ2 = Qe2 /C2 = 1020/945 = 1.08 Putting these values into Eqs. (4.53) and (4.54), we have (2)
(2)
(2)
A2 = (1 − ρ2 ) · C2 · T2 + 1 − (1) = = (1 − 1.08) · 945/3600 · 600 + 1 − 19.7 = −31.30 (2)
(2)
(2)
B2 = 4 · ((1) + ρ2 · C2 · T2 ) = = 4 · (19.7 + 1.08 · 945/3600 · 600) = 759.20 and, therefore, with Eq. (4.52), we have
>(2)
= =
1 2 1 2
·
·
(2) (A2 )2
(2) + B2
(2) − A2
=
( − 31.30)2 + 759.20 + 31.30 = 36.5 veh
128
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
By specifying Eqs. (4.56) and (4.57), we have (2)
J2 =
T2 2
(2)
· (1 − ρ2 ) −
1 (2) C2
· ( < Ls2 > (1) + 1) =
600 1 2 · (1$ − 1.08) − 945/3600 · (19.7 + %1) = −102.86 (2) (2) (2) M2 = 4(2) · T22 · (1 − ρ2 ) + 12 · ρ2 · T2 = C2 $ % 4 1 · 600 · (1 − 1.08) + · 1.08 · 600 = 4571.43 = 945/3600 2 2
=
and therefore, from Eq. (4.55), we have (2)
<ws2 >=
1 2
=
· 1 2
(2) (2) (2) (J2 )2 + M2 − J2 = · (−102.86)2 + 4571.43 + 102.86 = 113.0 s
In step 3, we calculate (Eq. (4.4)), for each entry, the increase in demand caused by the number of vehicles in the system relative to the previous state; to do so, the (2) must be converted from veh to pcu, once again using the coefficient f = 0.962. (3) Therefore, for entry 2, since Qe2 = 1019 pcu/h = 1019 0.962 = 980 veh/h and (2) = 36.5/0.962 = 37.94 pcu/h, we have (3)*
Qe2 = 1019 + 37.94 ·
60 = 1247 pcu/h 10
Since we know the demand [Qei ∗ ], we determine, for each entry, the disturbing (3) flows and then the capacity values [Ci ] relative to calculation step 3, that is, for the interval T3 . (3) (3) For C2 our calculation yields C2 = 1040 pcu/h (other two entries are oversaturated, and, thus, for the calculation of capacity, we used the computational procedure described in Sect. 2.5). (3) To determine <ws2 > (3) and (3) , we convert C2 from pcu/h to veh/h. Thus, we have (3)
(3)
C2 = 1040 · 0.962 = 1000 veh/h and we evaluate traffic intensity during the period T3 as: (3)
(3)
(3)
ρ2 = Qe2 /C2 = 980/1000 = 0.98 Putting these values into Eqs. (4.53) and (4.54), we have (3)
(3)
(3)
A2 = (1 − ρ2 ) · C2 · T3 + 1 − (2) = = (1 − 0.98) · 1000/3600 · 600 + 1 − 36.5 = −32.17 (3) (3) (3) B2 = 4 · ((2) + ρ2 · C2 · T3 ) = = 4 · (36.5 + 0.98 · 1000/3600 · 600) = 799.33
4.2
State Evolution with Saturated or Oversaturated Entries
129
and, therefore, with Eq. (4.52), we have
(3)
= 12
·
= 12 ·
(3) (A2 )2
(3) (3) + B2 −A2
=
( − 32.17)2 + 799.33 + 32.17 = 37.5 veh
By specifying Eqs. (4.56) and (4.57), we have (3)
J2 =
T3 2
(3)
· (1 − ρ2 ) −
· ( (2) + 1) =
600 1 2 · (1$ − 0.98) − 1000/3600 · (36.5 + % 1) = −129.00 (3) (3) T3 4 = (3) · 2 · (1−ρ2 ) + 12 · ρ2 · T3 = C2 $ % 4 1 · 600 = 1000/3600 2 · (1 − 0.98) + 2 · 0.98 · 600 = 4320.00
= (3) M2
1 (3) C2
and, therefore, from Eq. (4.55), we have (3) <ws2
1 2
>=
·
=
1 2
·
(3) (J2 )2
(3) + M2
(3) − J2
=
(−129.00)2 + 4320.00 + 129.00 = 136.9 s
In step 4, we calculate (Eq. (4.4)), for each entry, the increase in demand caused by the number of vehicles in the system relative to the previous state; to do so, the (3) must be converted from veh to pcu, once again using the coefficient f = 0.962. (4) Thus, for entry 2, since Qe2 = 988 pcu/h = 988 · 0.962 = 950 veh/h and (3) = 37.5/0.962 = 38.98 pcu/h, we have (4)∗
Qe2 = 988 + 38.98 ·
60 = 1222 pcu/h 10
Since we know the demand [Qei ∗ ], we determine, for each entry, the disturbing (4) flows and then the capacity values [Ci ] relative to calculation step 4, that is, for the interval T4 . (4) (4) For C2 , our calculation yields C2 = 1,227 pcu/h (other two entries are oversaturated, and, thus, for the calculation of capacity, we used the computational procedure described in Sect. 2.5.) (4) To determine <ws2 >(4) and (4) , we convert C2 from pcu/h to veh/h. Thus, we have (4)
(4)
C2 = 1227 · 0.962 = 1180 veh/h
130
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
and we evaluate traffic intensity during the period T4 as: (4)
(4)
(4)
ρ2 = Qe2 /C2 = 950/1180 = 0.81 Putting these values into Eqs. (4.53) and (4.54), we have (4)
(4)
(4)
A2 = (1 − ρ2 ) · C2 · T4 + 1− < Ls2 >(3) = = (1 − 0.81) · 1180/3600 · 600 + 1 − 37.5 = 0.87 (4) (4) (4) B2 = 4 · ((3) +ρ2 · C2 · T4 ) = 4 · (37.5+ 0.81 · 1180/3600 · 600) = 787.20 and, therefore, with Eq. (4.52), we have (4) = =
·
1 2 1 2
(4) (4) (4) (A2 )2 + B2 − A2 = · (0.87)2 + 787.20 − 0.87 = 13.6 veh
By specifying Eqs. (4.56) and (4.57), we have (4)
J2 =
T4 2
(4)
· (1 − ρ2 ) −
1 (4) C2
· ( < Ls2 > (3) + 1) =
600 1 1) = −60.46 2 · (1$ − 0.81) − 1180/3600 · (37.5 + % (4) (4) (4) T4 4 1 M2 = (4) · 2 · (1 − ρ2 ) + 2 · ρ2 · T4 = C2 $ % 4 1 · 600 = 1180/3600 2 · (1 - 0.81) + 2 · 0.81 · 600 = 3361.02
=
and, therefore, from Eq. (4.55), we have (4) <ws2
> = =
1 2
(4) (J2 )2
(4) M2
(4) − J2
· + = · (-60.46)2 + 3361.02 + 60.46 = 73.0 s 1 2
In step 5, we calculate (Eq. (4.4)), for each entry, the increase in demand caused by the number of vehicles in the system relative to the previous state; to do so, the < Lsi > (4) must be converted from veh to pcu, once again using the coefficient f = 0.962. (5) Thus, for entry 2, since Qe2 = 956 pcu/h = 988 · 0.962 = 920 veh/h and (4) = 13.6/0.962 = 14.14 pcu/h, we have (5)*
Qe2 = 956 + 14.14 ·
60 = 1041 pcu/h 10
Since we know the demand [Qei ∗ ], we determine, for each entry, the conflicting (5) flows and then the capacity values [Ci ] relative to calculation step 5, that is, for the interval T5 . (5) (5) For C2 our calculation yields C2 = 1247 pcu/h (other three entries are oversaturated, and, thus, for the calculation of capacity, we used the computational procedure described in Sect. 2.5). (5)
4.2
State Evolution with Saturated or Oversaturated Entries
131 (5)
To determine < ws2 > (5) and < Ls2 > (5) , we must convert C2 from pcu/h to veh/h. Thus we have (5)
C2 = 1247 · 0.962 = 1200 veh/h and we evaluate traffic intensity during the period T4 as: (5)
(5)
(5)
ρ2 = Qe2 /C2 = 920/1200 = 0.77 Putting these values into Eqs. (4.53) and (4.54), we have (5)
(5)
(5)
A2 = (1 − ρ2 ) · C2 · T5 + 1 − (4) = = (1 − 0.77) · 1200/3600 · 600 + 1 − 13.6 = 33.40 (5) (5) (5) B2 = 4 · ( (4) + ρ2 · C2 · T5 ) = 4 · (13.6 + 0.77 · 1200/3600 · 600) = 670.40 and, therefore, with Eq. (4.52), we have < Ls2 >(5) =
1 2
=
· 1 2
(5) (5) (5) (A2 )2 + B2 −A2 = · (33.40)2 + 670.40 − 33.40 = 4.4 veh
By specifying Eqs. (4.56) and (4.57), we have (5)
J2 =
T5 2
(5)
· (1 − ρ2 ) −
· ( (4) + 1) =
600 1 2 · (1$ − 0.77)− 1200/3600 · (13.6 + 1) % = 25.20 (5) (5) T5 4 = (5) · 2 · (1 − ρ2 ) + 12 · ρ2 · T5 = C2 $ % 4 1 · 600 = 1200/3600 2 · (1 − 0.77) + 2 · 0.77 · 600 =
= (5) M2
1 (5) C2
3600.00
and, therefore, from Eq. (4.55), we have (5)
<ws2 > =
1 2
·
=
1 2
(5) (5) (5) (J2 )2 + M2 − J2 = · (25.20)2 + 3600.00 − 25.20 = 19.9 s
The results of this worked example are shown in Fig. 4.3, which shows that, by the procedure adopted, it is possible to follow the behavior in time of the development of the queue (increase and clearance) and of the time spent in the system.
132
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
4.2.2 Second Worked Example If traffic demand is given as a time continuous function, the procedure described in Sect. 4.2.1, implemented with the calculation steps Tk of reduced value (e.g., 1 min), allows us to regularly follow the evolution of the system and to obtain the description of the behavior of the intersection under general service conditions. As an example of what we have just described, using the calculation code R , where the above‘Roundabout’ developed by M. Corradini in Visual Basic mentioned procedure is implemented, the case of a four-legged roundabout was analyzed for traffic demands that have different behaviors for all of the entries. For the first and second legs, the demand is parabolic; for the third and fourth legs, the demand is sinusoidal; and the instants of beginning and end of the peaks and their duration are different for each entry (See Fig. 4.4). In addition, the time variation of matrix PO/D of the assumed traffic percentages is such that entries 2 and 4 are, during the evolution of the system, always undersaturated (See Fig. 4.5). Figure 4.4 also shows the relationship Ci = Ci (t) obtained with the capacity calculation performed for the scheme under examination. With the examination of the behavior of the average number of users and average times spent in the system, described in Figs. 4.6 and 4.7, respectively, it is possible to follow all aspects of the evolution of the roundabout. Therefore, we can evaluate, in particular, the effects that the time variation of traffic demand and, consequently of capacity, have on the waiting phenomena that occur at each entry.
Fig. 4.4 Traffic demand and capacity versus time at the four legs of a roundabout
4.2
State Evolution with Saturated or Oversaturated Entries
133
Fig. 4.5 Traffic intensity (degree of saturation) versus time at the four legs of a roundabout
Fig. 4.6 The average number of users in the system versus time at the four legs of a roundabout
134
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
Fig. 4.7 The average time spent in the system versus time at the four legs of a roundabout
4.3 Evaluation of Levels of Service For the evaluation of Levels of Service (LOS) of at-grade unsignalized intersections, we generally follow the indications of the American Highway Capacity Manual (HCM) [2]. The Manual distinguishes six LOS, from A to F, based on quality levels of circulation at the intersection, in descending order (A excellent; F very poor). Among the variations introduced in the successive editions of the HCM, there is also a different selection of the parameters relative to Levels of Service. In the edition published in 1985, the statement of LOS is given on the basis of intervals of reserve capacity RC = Ci – Qei (See Table 4.1), and it gives only qualitative indications about delays.
Table 4.1 LOS, reserve capacity, and average delay according to HCM 1985 LOS
Reserve capacity
Average delay
A B C D E F
>400 300–400 200–300 100–200 0–100 <0
zero or short short medium high very high –
4.3
Evaluation of Levels of Service
135
Table 4.2 LOS and average time in the system according to HCM 1994–HCM 1997 and HCM 2000 HCM 94–HCM 97
HCM 2000
LOS
Average delay (s)
Average delay (s)
Average time spent in the system (s)
A B C D E F
<5 5–10 10–20 20–30 30–40 >45
<10 10–15 15–25 25–35 35–50 >50
<5 5–10 10–20 20–30 30–40 >45
In the editions published in 1994, 1997, and 2000, the Levels of Service are based on the levels of acceptability of the waits by the users, according to the values shown in Table 4.2. Regarding the parameters shown in Table 4.2, we recall that “delay” means the time w defined with Eq. (1.21) of Chap. 1. In Table 4.2, according to HCM 2000, it is assumed to be systematically greater than 5 s of the time spent in the system ws . It is worth noting that the classifications of LOS on the basis of reserve capacity RC or the waiting time are the same. In fact, it is possible to demonstrate [3] that if we have a calculation procedure of the time spent in the system, we can identify a bijective relationship between the two parameters assumed to represent the Level of Service, and we can use that relationship to deduce the value of one of the parameters after the value of the other parameter is known. Thus, for example, if the system is under steady-state conditions with Poissonian arrivals and exponential waiting times, from Eqs. (3.17) and (3.3) of Chap. 3, we immediately have, E[ws ] =
1 1 1 = = C · (1 − ρ) C − Qe RC
(4.58)
With Eq. (4.58), on the basis of the intervals of reserve capacity shown in Table 4.1, we have the intervals of waiting times shown in Table 4.3. For RC < 0, Eq. (4.58) loses significance (See Fig. 3.3 of Chap. 3) consistently with the circumstance that, for RC ≤ 0, there are no statistical equilibrium conditions for the system. Table 4.4 shows the correspondence calculated under steady-state conditions (RC > 0) in [4] between average times spent in the system E[ws ] and reserve capacity, starting with various formulas for the evaluation of E[ws ] and with reference to the intervals of reserve capacity shown in Table 4.1.
136
4
Calculation of Waiting Times, Queue Lengths, and Levels of Service
Table 4.3 LOS, reserve capacity, and average time spent in the system according to Eq. (4.58) LOS
Reserve capacity
Average time spent in the system
A B C D E F
<400 300–400 200–300 100–200 0–100 <0
<9 9–12 12–18 18–36 >36 –
Table 4.4 Reserve capacity and average time spent in the system according to various researchers Average time spent in the system (s)
LOS
Reserve capacity
According to Kremser
According to Brilon and Grossman
According to Siegloch
According to Mauro
A B C D E F
<400 300–400 200–300 100–200 0–100 <0
<10 10–12 12–15 15–20 >35 –
<10 10–15 15–25 25–45 >45 –
<5 5–9 9–15 15–30 >30 –
<10 10–15 15–22 22–44 >44 –
With the absence of steady-state conditions, because of saturation or oversaturation of the entries (RC ≥ 0) and/or because of the short duration of the period Tk of demand variation and/or capacity, a correspondence between reserve capacity and average times spent in the system may be obtained by expressing the time-dependent solutions as a function of RC instead of traffic intensity (degree of saturation) ρ. Working in this way, it is possible to demonstrate that a reserve capacity RC greater than 100 veh/h is associated with a time spent in the system smaller than 40 s. The threshold of 45 s given by the HCM 2000 for LOS E for at-grade unsignalized intersections (See Table 4.2) is reached for values of RC of 75–80 veh/h (See Fig. 4.8, [5]). Anyway, under any condition of the system (steady-state or transient) in today’s technical practice, in absence of specific alternative indications, we generally use the classification elaborated by HCM 2000 for linear, unsignalized intersections. According to this classification, the quality of the circulation at the intersection is established on the basis of the level of acceptability by the users of the waits at entries. Specific studies of roundabouts have not been available. Therefore, the only solution is to use the criterion of HCM 2000 described in Table 4.2 for roundabouts.
References
137
Fig. 4.8 Reserve capacity, time spent in the system, and levels of service [5]
The evaluation of LOS for roundabouts is performed by assuming that the roundabout entry that is characterized by the worst LOS determines the LOS of the entire intersection. Finally, it is well known that short times are generally associated with transient conditions, so it is worth noting that the recording of the short times when there are poor Levels of Service does not invalidate the suitability evaluation of the intersection to perform its functions. When this occurs, we must accurately evaluate the singular, actual situations on a case-to-case basis.
References 1. Brilon, W., Troutbeck, R., Tracz, M., Review of International Practices Used to Evaluate Unsignalized Intersections, Transportation Research Board Circular, number 468, Washington, April 1997. 2. Transportation Research Board, Highway Capacity Manual, Washington DC, edition 1985, 1994, 1997, 2000. 3. Brilon, W. (Editor), Intersections Without Traffic Signals, Vol. I, Springer-Verlag, Berlin, 1988. 4. Mauro, R., Contributo alla progettazione delle intersezioni stradali non semaforizzate. [Contribution to the design of unsignalized road intersections] (in Italian), Report of D.I.T. n◦ 39/95, Naples, 1995. 5. Brilon, W., Kreisel – Handbuch, Anhang A, Version Oktober 2004.
Chapter 5
Evaluation of Roundabout Reliability
The study of the performances of roundabouts in terms of capacity indices (See Sect. 1.2) may be completed using further analysis in addition to the ones that we presented in the previous chapters. This occurs because the flows of the various legs (and the capacities that depend on them) are random variables. In general, they are non-statistically independent. Therefore, to evaluate reliability, that is to say the probability that the system does not fail (in the specific case, that demand does not exceed the single entry capacity) it is necessary to characterize the flows and their related values by means of their probability functions or when these laws are not available, by coincise indices such as means, variances, and covariances. This chapter presents general criteria for the evaluation of reliability in each leg based on the study of the performance function Z (Z = C – Qe or Z = C/Qe ) and it provides the analytical relations in the particular case in which capacity and demand (and thus also Z) are normally distributed with means and variances known. An approximated criterion is also provided to be used in cases in which the probability laws of capacities and demands, and thus of performance function Z, are unknown or difficult to determine. The worked examples developed to illustrate the method (and the many others that have not been reported for the sake of brevity) show, as it was logical to expect, that the two only indices normally used (Reserve Capacity and/or Rate of Capacity) are not sufficient to ensure that the system does not fail. When the mean of the Reserve Capacity and/or of the Capacity Rate does not change, reliability depends on the level of uncertainty that affects the values in question (dispersion around the mean values of the flows). Regarding the threshold value to attribute to reliability, it must be stated that it cannot be fixed in general terms, but it should be identified on a case-to case basis in relation to the damage (excessive mean and global waiting times, safety decrease, repercussions on the surrounding network) caused by the system failure. The results presented in this chapter can help to analyze roundabouts in a better and more rational way. Finally, even other performance indices for roundabouts, as simple and total capacity, can be expressed in a probabilistic way.
R. Mauro, Calculation of Roundabouts, DOI 10.1007/978-3-642-04551-6_5, C Springer-Verlag Berlin Heidelberg 2010
139
140
5 Evaluation of Roundabout Reliability
5.1 Reliability and Performance Functions Consider a roundabout entry of capacity C, interested by an entering traffic demand Qe . To evaluate the reliability – intended as the value “Z” of the suitability of a system to perform its function – it is natural to compare the values C and Qe . The “Z” value, that is to say “performance function”, can be measured both with the difference C – Qe and with the ratio C/Qe , which in the reliability theory terminology [1] are indicated as reliability margin and reliability factor respectively: Z = C − Qe
(5.1)
Z = C/Qe
(5.2)
Equations (5.1) and (5.2) are connected to the two most widely adopted roundabout capacity indices used to characterize service conditions: in fact, Eq. (5.1) coincides with the reserve capacity (RC), whereas Eq. (5.2) is the reciprocal of the rate of capacity when it is expressed in absolute terms (as we said in the previous Sect. 1.2). Therefore, once prefixed two given minimum values zmin and z min for the reliability margin and reliability factor, the reliability condition of the system can be written: Z = C − Qe > zmin
(5.3)
Z = C/Qe > zmin
(5.4)
Particularly, in the former value the limit zmin = 0 can be taken and in the latter z min =1, so that the success condition is represented by C − Qe > 0
(5.5)
C/Qe > 1
(5.6)
The complementary relations of Eq. (5.5) and Eq. (5.6) represent failure conditions: C − Qe < 0
(5.7)
C/Qe < 1
(5.8)
What has been expounded so far is the deterministic position of the problem. Because of the random nature of the factors and relations on which traffic capacity and traffic demand values depend, the previous values C and Qe can vary randomly.
5.1
Reliability and Performance Functions
141
Fig. 5.1 Probability density function (p.d.f.) examples of the performance function (random variables Z)
Therefore, we consider C and Qe as random variables described by given probability laws or by adequate statistics, the above-mentioned relations must be suitably modified to take this circumstance into account. Thus, the values “Z” that are yielded by Eq. (5.1) and by Eq. (5.2) are also to be considered, insofar as they are random variable functions, as random values. Also, it should be considered that each entering traffic demand value Qe at an entry of capacity C corresponds to a reliability value determination. Assume, then, in the more general case, that the probability law of the random variable “Z” (see Eqs. (5.1) and (5.2)) can be represented by the probability density function, p.d.f., fZ (See Fig. 5.1). On the basis of PZ probability, the fractiles zmin and z min can be determined: this equals to let PZ = P {Z = C − Qe ≤ zmin } = FZ (zmin )
(5.9)
1 0 PZ = P Z = C/Qe ≤ zmin = FZ (zmin )
(5.10)
where FZ (Z) are the distribution functions (c.d.f.) of the random variable “Z” in these two cases, Z = C – Qe and Z = C/Qe . PZ is then the probability of the event {entry unreliability}, given that zmin and z min are minimum values prefixed by the reliability value. The complementary event probability {entry reliability} obviously results in 1 − PZ = P {Z = C − Qe > zmin } = 1 − FZ (zmin )
(5.11)
1 0 1 − PZ = P Z = C/Qe > zmin = 1 − FZ (zmin )
(5.12)
The “failure” event and the complementary “success” event probabilities are then obtained respectively (See Fig. 5.2): Pf = P {Z = C − Qe ≤ 0} = FZ (0) Pf = P {Z = C/Qe ≤ 1} = FZ (1)
failure
(5.13)
142
5 Evaluation of Roundabout Reliability
Fig. 5.2 Examples of the identification of the probability of “failure” and “success” events
A = 1 − Pf = P {Z = C − Qe > 0} = 1 − FZ (0) success A = 1 − Pf = P {Z = C/Qe > 1} = 1 − FZ (1)
(5.14)
From now on, the probability 1 – Pf associated with the “success” event will be briefly indicated by reliability A. In particular, with reference to the performance variable Z = C – Qe , if fCQe (c,qe ) is the combined p.d.f of C and Qe , for Eq. (5.13) it results that: (5.15) Pf = P(Z ≤ 0) = fCQe (c,qe )dc dqe D
In Eq. (5.15) D is the unsafe region, where the performance function Z = C − Qe takes values Z ≤ 0 (See Fig. 5.3 for (C, Qe ) ≥ 0). In other words, according to Eq. (5.15), the volume subtended by fCQe (c,qe ) in correspondence with the region D gives the value Pf .
Fig. 5.3 Graphic identification of the unsafe region
5.2
General Calculation Procedure of Reliability
143
In conclusion, the problem of the evaluation of a roundabout reliability refers, in its general formulation, to the thorough probabilistic characterization of the values that the entry flows Qe can take and to the entry capacity C. However, as it will be demonstrated in the following discussion, when this thorough characterization is not available, Level 1 reliability methods can be performed on the basis of statistical estimations of the expected value and of the standard deviation of the random variables Qe and C, that is to say using only one measurement variability of the values in question (the standard deviations of the random √ sQe = VAR Qe and sC = VAR [C] , once the mean values E[Qe ] and E[C] are known).
5.2 General Calculation Procedure of Reliability Adopting Eq. (5.1), Z = C − Qe as a performance variable for a common entry, if C and Qe are random variables that are statistically independent, with the calculation rules for double integrals from Eq. (5.15) the two following equivalent expressions are obtained for Pf Pf = Pf =
+∞ # 0 +∞ #
fC (c)[1 − FQe (c)]dc (5.16) fQe (q)FC (q)dq
0
where fC (·) and fQe (·) are the p.d.f. respectively of C and Qe , and Fc (·) and FQe (·) are the c.d.f. respectively of C and Qe . To prove Eq. (5.16) we start from Figs. 5.4 and 5.5.
Fig. 5.4 Tails of p.d.f. of demand Qe and of capacity C (generic p.d.f.)
144
5 Evaluation of Roundabout Reliability
Fig. 5.5 Tails of p.d.f. of demand Qe and of capacity C (generic p.d.f.)
Figure 5.4 shows the tails of the p.d.f. of demand Qe and of capacity C (generic p.d.f.). The probability that demand Qe is bigger than an assigned value c of Capacity C is equal to P(Qe > c) = 1 − FQe (c) The probability that capacity C falls in the neighborhood of c is equal to P(c − dc < C ≤ c) = fc (c)dc The “failure” probability when C is equal to a given c is dPfc = [1 − FQe (c)] · fc (c)dc For all the possible c, it results 2+∞ Pf = [1 − FQe (c)] · fc (c)dc 0
which is Eq. (5.16). Dually (See Fig. 5.5) P(C ≤ q) = Fc (q) P(q < Qe ≤ q + dq) = fQe (q)dq dPfQe = fQe (q)Fc (q)dq
5.2
General Calculation Procedure of Reliability
145
For all the possible q it results 2+∞ fQe (q)Fc (q)dq Pf = 0
which is Eq. (5.16). From Eq. (5.16), if C and Qe are normally distributed, for Pf it results [1] 1 Pf = √ 2π
2−β
e−t
2 /2
dt = 0.5 − erf(β)
(5.17)
−∞
where erf(·) is the error function and β is the safety index (inverse of the coefficient of variation of the performance function Z) β = E[Z]/ VAR[Z]
(5.18)
√ with E[Z] and VAR [Z] as the expected value and the standard deviation of the performance function Z respectively (Eq. (5.1)). If C is normally distributed with mean C and variance σC2 and Qe is distributed, for example, exponentially with parameter α, for Pf it results that !
C Pf = F − σc
" + exp{
0.5α 2 σc2
!
C − αC} 1 − F − + ασ c σc
" (5.19)
where F(·) is the c.d.f. of the standardized normal distribution.
5.2.1 A Worked Example For this example, the capacity formulation of SETRA [2] is adopted. In Fig. 5.6 a roundabout with flow indications and geometrical features for the calculation of C capacity is schematically reported. For all legs we have: – entry width ENT = 4.00 m – splitter island SEP = 6.00 m The circulatory roadway width is ANN = 8.00 m. For an entry capacity “i” the formulation selected gives C = (1330 − 0.7 · Qdi ) · [1 + 0.1 · (ENT − 3.50)]
(5.20)
146
5 Evaluation of Roundabout Reliability
Fig. 5.6 Geometrical layout of the roundabout
where the disturbing flow Qdi is 2 Qdi = Qci + Qui [1 − 0.085 · (ANN − 8)] 3
(5.21)
Qci is the circulating flow in front of the leg considered, Qui is the traffic exiting the leg i selected Qui = Qui
15 − SEP 15
(5.22)
With the given values of ENT, SEP and ANN, Eqs. (5.20), (5.21) and (5.22) yield C = 1397 − 0.735 · Qdi
(5.23)
Qdi = Qci + 0.4 · Qui
(5.24)
with
5.2
General Calculation Procedure of Reliability
147
Table 5.1 Matrix O/D of the mean values E[Qij ] in (pcu/h) and variance VAR[Qij ] in (pcu/h)2 D
1
2
3
4
150 2000 100 1800 – – 200 2500 450 6300
250 2200 150 900 100 1800 – – 500 4900
Qei
O 1 2 3 4 Qui
– – 300 6400 300 3000 150 1500 750 10900
200 2500 – – 200 2500 350 9000 750 14000
600 6700 550 9100 600 7300 700 13000 – –
Supposing that, on the basis of experimental observations, mean flow E[Qij ] and variance VAR[Qij ] of all the turnings have been estimated and that these flows result in being statistically independent. These data are reported in the matrix O/D of Table 5.1. In Table 5.1 in each square ij the top value is the mean, and the bottom one is the variance associated with the turning from leg i into leg j. Table 5.1 also shows the estimated values of the mean and variance for the overall flows for each leg at entry Qei and at exit Qui . The flows are expressed in pcu/h. Mean and Variance of the Circulating Flows Qci The circulating flows Qc are linear functions of traffic demand expressed as turning flows. For example, for entry 1 it results in Qc1 = Q24 + Q23 + Q34 . On the basis of the properties of mean E[·] and variance VAR[·] operators, when applied to random variables statistically independent and with the data of Table 5.1, it gives E[Qc1 ] = E[Q24 ] + E[Q23 ] + E[Q34 ] = 150 + 100 + 100 = 350 pcu/h VAR[Qc1 ] = VAR[Q24 ] + VAR[Q23 ] + VAR[Q34 ] = = 900 + 1800 + 1800 = 4500 (pcu/h)2
(5.25) (5.26)
The coefficient of variation (cv)1 is (cv)1 =
VAR[Qc1 ]/E[Qc1 ] = 67/350 = 0.192
(5.27)
Doing the same for the other entries, the values of the mean and variance for the circulating flows in Table 5.2 can be obtained.
148
5 Evaluation of Roundabout Reliability
Table 5.2 Statistics of circulating flows, disturbing flows, and entry capacities. Safety index and reliability index values E[Qci ] (pcu/h)
E[Qdi ] (pcu/h)
E[Ci ] (pcu/h)
Average reserve capacity
VAR[Qci ] VAR[Qdi ] VAR[Ci ] (pcu/h)2 (pcu/h)2 (pcu/h)2
E[C]-E[Qe ] (pcu/h)
βi
A
1
350 4500
650 6244
920 3373
320
3.19
0.999
2
550 6300
850 8540
772 4613
222
1.90
0.971
3
700 13000
880 14008
750 7567
150
1.23
0.897
4
450 6300
650 7084
919 7084
219
1.55
0.939
Entry
Mean and Variance of disturbing flows Qdi For entry 1 Eq. (5.24) gives Qd1 = Qc1 + 0.4 · Qu1 where Qu1 = Q21 + Q31 + Q41 . On the basis of the above-mentioned properties of the mean and variance operators and with the values of Table 5.1, it results in: E[Qd1 ] = E[Qc1 ] + 0.4 · (E[Q21 ] + E[Q31 ] + E[Q41 ]) = = 350 + 0.4 · (300 + 300 + 150) = 650 pcu/h
(5.28)
2
VAR[Qd1 ] = VAR[Qc1 ] + 0.4 · (VAR[Q21 ] + VAR[Q31 ] + VAR[Q41 ]) = = 4500 + 0.16 · (6400 + 3000 + 1500) = 6244 (pcu/h)2 (5.29) In this case, the coefficient of variation equals to (cv)1 =
VAR[Qd1 ]/E[Qd1 ] = 79/650 = 0.122
(5.30)
Repeating the calculation for the other entries, the disturbing flow statistics reported in Table 5.2 can be determined. Mean and Variance of Flow Capacities Ci For entry 1, with Eq. (5.23), since the relation between Ci and Qdi is linear and with the above-mentioned values calculated for Qdi moments, it results that E[C1 ] = 1397 − 0.735E[Qd1 ] = = 1397 − 0.735 · 650 = 920 pcu/h
(5.31)
5.2
General Calculation Procedure of Reliability
149
2
VAR[C1 ] = 0.735 · 6244 = 3373 (pcu/h)2
(5.32)
(cv)1 = 58/920 = 0.063
(5.33)
For the other entries, the capacity moments reported in Table 5.2 can be determined in the same way. Reliability Calculation For entry 1, the performance function (5.1) Z1 is on average equal to E[Z1 ] = E[C1 ] − E[Qe1 ] = 920 − 600 = 320 pcu/h VAR[Z1 ] = VAR[C1 ] + VAR[Qe1 ] = 3373 + 6700 = 10073 (pcu/h)2 It follows that β1 = √
320 E[Z1 ] = 3.19 =√ VAR[Z1 ] 10073
and thus erf(β1 ) = erf(3.19) = 0.499. With Eq. (5.17) reliability A for entry 1 on the basis of Eq. (5.14) equals to A = 0.5 + erf(3.19) = 0.999. Table 5.1 shows the reliability values calculated for the remaining entries to the roundabout considered. The present example has been carried out using the hypotheses of mutual statistical independence among entering flows. If this circumstance does not occur, it is also necessary to take into consideration the covariance cov[(·);(·)] of statistically dependent flows as shown below. For example, with reference to entry 1, suppose that the elaboration of experimental data traffic shows the mutual statistical dependence among the circulating flows Q24 ; Q23 ; Q34 and among Q21 ; Q31 ; Q41 , so to obtain the values of Table 5.3 for the covariances (the Qij flows are expressed in pcu/h). Table 5.3 Values of the cov[(·);(·)] in (pcu/h)2 Qij
Q23
Q31
Q34
Q41
Qij Q21 Q31 Q23 Q24
3500
1000
2000 1800 1800 1100
150
5 Evaluation of Roundabout Reliability
While E[Qc1 ] remains equal to Eq. (5.25), with the values of Tables 5.1 and 5.3, the VAR[Qc1 ] is, with respect to Eq. (5.26), thus modified: VAR Qc1 = VAR Q24 + VAR Q23 + VAR Q34 + +2 cov Q24 ;Q23 + cov Q24 ;Q34 + cov Q23 ;Q34 = = 900 + 1800 + 1800 + 2 · (1000 + 1100 + 1800) = 12300(pcu/h)2 (5.26 ) For the disturbing flow Qd1 the same value yielded by Eq. (5.28) is obtained for the mean E[Qd1 ], while for the calculation of VAR[Qd1 ] it is necessary to know the cov[(Qc1 );(Qu1 )], as well as the covariances of Table 5.3. Suppose that for cov[(Qc1 );(Qu1 )] it results, on the basis of traffic measurement treatment, cov[(Qc1 );(Qu1 )]=13000 (pcu/h)2 . With the values of Tables 5.1 and 5.3, it is obtained: VAR Qu1 = VAR Q21 + VAR Q31 + VAR Q41 + +2 cov Q21 ;Q31 + cov Q21 ;Q41 + cov Q31 ;Q41 = (5.34) = 6400 + 3000 + 1500 + 2 · (3500 + 2000 + 1800) = 2 = 25500(pcu/h) 2
VAR[Qd1 ] = VAR[Qc1 ] + 0.4 · VAR[Qu1 ] + 2 · 0.4 · cov[Qc1 ;Qd1 ] = = 12300 + 0.16 · 25500 + 2 · 0.4 · 13000 = 26780 (pcu/h)2
(5.35)
With Eq. (5.35) and with Eq. (5.28), on the basis of Eq. (5.20), for entry 1 capacity it is obtained that: E[C1 ] = 1397 − 0.735E[Qd1 ] = 1397 − 0.735 · 650 = 920 pcu/h 2
2
VAR[C1 ] = 0.735 · VAR[Qd1 ] = 0.735 · 26780 = 14467 (pcu/h)2
(5.36) (5.37)
In the end, with E[Qe1 ]= 600 pcu/h (See Table 5.1) for the performance function Z1 it results: E[Z1 ] = E[C1 ] − E[Qe1 ] = 920 − 600 = 320 pcu/h VAR[Z1 ] = VAR[C1 ] + VAR[Q1 ] = 14467 + 6700 = 21167 (pcu/h)2
(5.38) (5.39)
Thus 320 E[Z1 ] = β1 = √ = 2.20 145.5 VAR[Z1 ]
(5.40)
and for reliability A it results (See Eqs. (5.14) and (5.17)) A = 0.5 + erf(β) = 0.5 + erf(2.20) = 0.986.
(5.41)
5.3
Approximated Calculation Procedure of Reliability
151
5.3 Approximated Calculation Procedure of Reliability This procedure is exclusively based on the knowledge of the mean value and on only one measurement of the random variability (generally, variance or standard deviation) of the values involved: it can thus be considered an approximated approach to evaluate reliability compared to the criterion illustrated in the previous section. Bearing in mind the meaning and the definition of mean value and variance, this method could be called the two-moment method. The use of this approximated method requires the introduction of a safety index β which can be defined using the performance function Z. This index is the number of standard deviations that separate the mean value Z from the value Z = 0 which – by definition – corresponds to the failure limit. This method is structured as follows. Calculate the mean and variance statistics E[Z] and VAR[Z] of the performance variable (5.1) Z = C − Qe
(5.1)
starting from the known homologues of Qe and of C (E[Qe ]; VAR[Qe ]; [E[C];VAR[C]) with the relation (See Fig. 5.7) E[Z] − βsz = 0
(5.42)
index β is calculated, and it provides the number of standard deviations sZ = √ VAR [Z] of Z that separate the mean value E[Z] from the value Z=0, corresponding by definition to the limit that marks the failure condition (Z = 0 ↔ C = Qe ∀ Qe ;C = 0).
Fig. 5.7 Reliability index b and p.d.f. of the performance function Z
152
5 Evaluation of Roundabout Reliability
By introducing the normalized performance function as ξ=
Z − E[Z] sz
(5.43)
the “success” and “failure” events are, respectively, equivalent to the occurrence of the inequalities: – success ξ ≥ −β
(5.44)
ξ < −β
(5.45)
– failure
In fact, putting in Eq. (5.13) a value of ξ ≥ −β yields Z, Z ≥ 0, that is to say that C ≥ Qe , while, substituting ξ < −β yields Z < 0, which equals to C < Qe . If the Z probability law, that is ξ, is known, each limit of β corresponds to a well-determined value of the failure probability Pf = P(ξ < −β = F( − β)
(5.46)
A = 1 − Pf = 1 − F( − β)
(5.47)
Even though the law distribution of the performance function Z is not known or easily determinable, β can be considered as a coherent reliability value. In fact, the tail of the most common probability density functions can be adequately approximated with an exponential function (suitably identified with two parameters A and b, (A>0, b>0), F(ξ) = A · exp(b · ξ)
if
F(ξ) =<< 1
(5.48)
For example, if Qe and C are both lognormally distributed, it can be demonstrated that Eq. (5.48) becomes F( − β) = 460 · exp( − 4.3β)
(5.49)
A = 1 − F( − β) = 1 − 460 · exp( − 4.3 · β)
(5.50)
and, thus,
It follows that as β increases, reliability also increases. With the most frequent f.d.p. the values of β at least equal to 2 always indicate quite high probabilities that Qe is systematically smaller than C, that is to say that the entry does not become saturated.
5.3
Approximated Calculation Procedure of Reliability
153
5.3.1 A Worked Example Suppose we adopt as a capacity formulation the one presented in the 2000 edition of the HCM [3] on the basis of which C=
Qc exp {−Qc Tc /3600} (pcu/h) 1 − exp {−Qc Tf /3600}
(5.51)
For an assigned value of Qc it results C = C(Tc ;Tf )
(5.52)
where Tc is the critical gap (s) and Tf is the follow-up time (s) (See Sect. 1.2). We assume that Qc is known without doubt and that Tc and Tf are instead random variables. Using the linearization of Eq. (5.51) it can be demonstrated [4] that an approximated evaluation of the first order of E[C] and VAR[C] is yielded by E[C] = C(E[Tc ];E[Tf ]) = Qc ·
exp{− Qc E[Tc ]/3600} (pcu/h) 1 − exp{− Qc E[Tf ]/3600}
VAR[C] = k21 VAR[Tc ] + k22 VAR[Tf ] + 2 · k1 · k2 · cov[Tc ;Tf ]
(5.53) (5.54)
where E[Tc ]; E[Tf ]; VAR[Tc ]; VAR[Tf ]; cov[Tc ;Tf ] have the usual meaning and for k1 and k2 it results k1 = k2 =
∂C ∂Tc ∂C ∂Tf
=−
Q2c exp{− Qc E[Tc ]/3600} · 3600 1 − exp{− Qc E[Tf ]/3600}
(5.55)
=−
Q2c exp{− Qc (E[Tc ] - E[Tf ])/3600} · 3600 (1 − exp{− Qc E[Tf ]/3600})2
(5.56)
E[Tc ];E[Tf ]
E[Tc ];E[Tf ]
Assume therefore, for an entry for which Qc = 250 pcu/h, the following values of the statistics of the random variables Tc and Tf : E[Tc ] = 4.4 s
VAR[Tc ] = 0.36 s2
E[Tf ] = 2.9 s
VAR[Tf ] = 0.25 s2
cov[Tc ; Tf ] = 0.30 s2
With these values and with C=C(Tc ,Tf ) given by Eq. (5.51), it is obtained for Eqs. (5.53), (5.54), (5.55) and (5.56)
154
k1 =
k2 =
5 Evaluation of Roundabout Reliability
250 exp{− 250 · 4.4/3600} =− · = −70 3600 1 − exp{− 250 · 2.9/3600}
250 exp{− 250 · (4.4 − 2.9)/3600} =− = −470 · 3600 (1 − exp{− 250 · 2.9/3600})2
2
∂C ∂Tc E[T ];E[T ] c f
2
∂C ∂Tf E[T ];E[T ] c f
E[C] = 250 ·
exp{− 250 · 4.4/3600} = 1010 pcu/h 1 − exp{− 250 · 2.9/3600}
VAR[C] = ( − 70)2 · 0.36 + ( − 470)2 · 0.25 + 2 · 0.30 · 32900 = 76729 (pcu/h)2 √ VAR[C] = 277 pcu/h The coefficient of variation (cv) is equal, in this case, to (cv) =
VAR[C]/E[C] = 277/1010 = 0.27
If E[Qe ] = 450 pcu/h and VAR[Qc ] = 1500 (pcu/h)2 it is obtained for the statistics of the performance function (5.1) E[Z] = E[C] − E[Qe ] = 1010 − 450 = 560 pcu/h √ √ VAR[Z] = VAR[C] + VAR[Qe ] = 76729 + 1500 = 280 pcu/h and thus, for β it results β=
560 = 2.00 280
This value of β means high reliability values. When it seems right to apply the approximation provided by Eq. (5.49) to this case the value (Eq. (5.50)) obtained for A is, for example, A = 1 − 460 exp {−4.3 · 2.00} = 0.92
5.4 Some Remarks When the mean of the Reserve Capacity and/or of the Capacity Rate does not change, reliability depends, on the level of uncertainty that affects the values in question (dispersion around the mean values of the flows). The role of the dispersions centered on the mean values E[Qe ] and E[C] is evident from the observation of the curves Pf = Pf (ωo ) in Fig. 5.8. They can be obtained as follows.
5.4
Some Remarks
155
Fig. 5.8 Pf = Pf (ωo ) for the couples of values ((cv)c ; (cv)Qe )
Expressed β as (Eq. (5.18)) ωo − 1 E[C] − E[Qe ] = β= VAR[C] + VAR[Qe ] ωo2 (cv)2c + (cv)2Qe √ where (cv)c = VAR [C]/E [C] and (cv)Qe = VAR Qe /E Qe are, respectively, the capacity and demand coefficients of variation at an entry and ωo = E[C]/ E[Qe ] the ratio among the means of the same ones, for Eq. (5.18) it results in Pf = Pf (ωo ;(cv)c ;(cv)Qe ) Once fixed the values that form the couples ((cv)c ; (cv)Qe ) of the table, with them, from Eq. (5.17), the 16 curves Pf = Pf (ωo ) of Fig. 5.8 can be obtained.
156
5 Evaluation of Roundabout Reliability
Figure 5.8 shows that: – for high values of (cv)c , even increasing considerably ωo , it is not possible to keep the failure probability within small values; – for small values of (cv)c , the variability of Qe is significant (this is instead unimportant for big values of (cv)c , that is to say with uncertain capacities).
References 1. Kottegoda, N., Rosso, R., Statistic Probability and Reliability for Civil and Environmental Engineers, MacGraw-Hill, New York, 1997. 2. SETRA/CSTR, Capacité des carrefours giratoires interurbains, Note d’information n◦ 44, Paris, 1987. 3. Highway Capacity Manual HCM 2000, Special Report n. 209, T.R.B., Washington DC, 2000. 4. Benjamin, R., Cornell, C. A., Probability, Statistics and Decision for Civil Engineers, MacGraw-Hill, New York, 1970.
Index
Note: Locators in boldface refer to Definitions of entries. A Acceleration phase, 11 Average Length of queue, see number of users in the queue number of users (vehicles) in the queue, 13 See also State and Queue number of users (vehicles) in the system, 13 See also State and Queue time spent in the system, 13 See also State and Queue waiting time in the queue, 13 See also State and Queue B Bondzio, 17 Bovy, 19 Brilon, 32, 42, 136 C Capacity, 7–9, 15–56 Brilon-Bondzio formula, 17–19 calculation at non steady-state conditions, 59–105 calculation at steady-state conditions, 17–36 exit and circle, 36 exit in presence of pedestrian crosswalks, 44 French formula, 27–32 German formula, 32–34 HCM 2000 formula (USA), 34–36, 153–154 in presence of pedestrian crosswalks Brilon, Stuwe and Drews formula, 42 CETE de l’Ouest formula, 41–43 Marlow and Maycock formula, 37–40
indices, 7–9, 21 rate, 9, 21 at entries (CRUe ), 21 at the conflicting points, 21 reserve (RC) (absolute and percentage), 8–9 simple, 9, 52–56 SETRA formula, 145–147 Swiss formula, 19–22 total, 9, 52, 54–56 United Kingdom formula, 22–27 CETE de l’Ouest, 41–43 Circulatory roadway Capacity, 36 number of lanes, 8, 15–16, 21, 27, 32–34 Circle, see Circulatory roadway Cumulative arrival count, 67–70 departure count, 67–70 D Deceleration phase, 11 Degree of saturation, 60–61 Delay, 1 geometric, 11 total, 11 Drews, 37, 41 E Entry oversaturation, 46–52, 121–134 See also Queue, deterministic analysis, time-dependent analysis (solution) Saturation, 46–52, 121–134 See also Queue, deterministic analysis, time-dependent analysis (solution) undersaturation, 107–121
R. Mauro, Calculation of Roundabouts, DOI 10.1007/978-3-642-04551-6, C Springer-Verlag Berlin Heidelberg 2010
157
158 See also Queue, probabilistic analysis Exit Capacity, 44 Erlang Law, 62 parameter k, 62 Exponential Law, 64, 65 F Flow circulating, 6, 7, 8 conflicting, 21 disturbing, 8 entering, 6, 8 exiting, 6, 8 pedestrian, 36–43 G Gap acceptance theory, 8 critical, 8, 32–35 Geometry of roundabout configuration, 8, 44 French capacity formula, 28 Standards, see configuration Swiss capacity formula, 20–21 TRRL capacity formula (United Kingdom), 23 GIRABASE, 27–32 Grossman, 136 Guichet, 27 H Highway Capacity Manual, 34–35, 134–136, 153–154 Hollis, 100, 102–103 K Kimber, 22, 100, 102–103 Kremser, 136 L Level of Service (LOS), 134–137 M Marlow and Maycock, 37 Mauro, 136 Measure of Effectiveness (MOE), 1 O O/D matrix (origin/destination matrix), 3 See also Traffic, demand P Passenger car unit, 16, 21 Peak period, 74–85, 98–104
Index Peak hour factor (phf), 4 Peak-hour volume, 4 Percentiles, 12, 113–114 PO/D , 3 See also Traffic, demand Poisson Law, 64–65 Pollaczeck-Khinchine relationship, 62 Q Queue deterministic analysis, 67–88 generalized time-dependent analysis (solution), 97–98 length, 1, 9–13 probabilistic analysis, 60–67 time-dependent analysis (solution), 88–105 R Rate, see Capacity Reliability calculation, 143–154 performance functions, 140–143 Reserve capacity (RC), see Capacity S Siegloch, 136 State conditions non steady-state, 2–4, 45–56, 88–105 steady-state, 2–4, 15–16, 60–67 evolution with saturated or oversaturated entries, 121–134 with undersaturated entries, 107–121 Stuwe, 37, 41 T Time follow-up, 8, 32–35 psychotechnical, 8 service, 10–11, 60–65, 70, 82–87 spent in the system, 9 See also State; Queue waiting in the queue, 10 See also State; Queue Traffic demand, 4–7 intensity, see Degree of saturation peak, see Peak period W Waiting time, 1, 9–13 See also State; Queue Waiting phenomena, see State; Queue
Index Wu, 16, 32, 38, 103 Worked examples on approximated calculation procedure of reliability, 153–154 on Bovy et al. formula (Switzerland), 21–22 on Brilon-Bondzio formula (Germany), 18–19 on Brilon-Wu formula (Germany), 32–34 on calculation of simple and total capacity, 52–56 on capacity calculation at saturation or oversaturation of entries, 46–52 on deterministic analysis of queues, 70–74, 77, 83–85
159 on general calculation procedure of reliability, 143–150 on GIRABASE formula (France), 29–31 on time-dependent analysis of queues, 91–94–96 on traffic demand at a roundabout, 4–7 on TRRL formula (United Kingdom), 25–27 on state evolution of roundabout with undersaturated entries, 111–121 on state evolution of roundabout with saturated or oversaturated entries, 123–134
